Some eastern Australian Acacia species, e.g., Acacia baileyana and A. mearnsii, are invasive in Western Australia. See:  Hussey B.M.J., Keighery G.J., Dodd J., Lloyd S.G. & Cousens R.D. (2007) Western weeds. A guide to the weeds of Western Australia. (available online).

Marsh frogs, Pelophylax (=Rana) ridibundus from eastern Europe are now established invasives in Britain, Spain and elsewhere in western Europe (see Kuzmin et al, 2009. "Pelophylax ridibundus". IUCN Red List of Threatened Species). Alpine newt Ichthyosaura alpestris is also established outside its natural European range (e.g. Beebee, T and Griffiths, R 2000 “Amphibians and reptiles: A Natural History of the British Herpetofauna”. HarperCollins. Also Ambystoma tigrinum is widely established outside its native range in N. America. For reptiles, we have, for example, the lizards Podarcis muralis and Lacerta bilineata, both introduced from mainland Europe to Britain and slowly spreading Deichsel, G., Gleed-Owen, C.P. & Mayer, W. (2007). Herpetol. Rev. 38 (1),100-101

I should also mention fish: Miro and Ventura (see attached).

Article Historical use, fishing management and lake characteristics ...

Article Evidence of exotic trout mediated minnow invasion in Pyrenea...

Truman, you are getting a lot of animals examples. and can add bullfrogs in western North America, rainbow trout successfully moved all over North America, mockingbirds and white-tailed deer moving across the Great Plains using tree-planted towns as stepping stones. Can't think of a single plant example at the moment but it is hard to imagine there are none. I'll think about ti some more.

I believe black locust Robinia pseudoacacia fits the bill. It is native to the southeastern US but is considered invasive in the upper Midwest and elsewhere.

Signal crayfish from the Pacific northwest, introduced throughout California (and Europe). Lots of other crayfish examples within North America. I'm not much of a plant person, but Spartina altiniflora from the Atlantic to the Pacific coast, fanwort (Cabomba caroliniana) from the southeast to the west. I suspect there may be more aquatic than terrestrial examples.

Gleditsia triacanthos is a tree native from USA that have been introduced throughout the American Continent. 

One example - Pittosporum undulatum is an Australian native species that is a serious environmental weed in remnant forest near Melbourne, Australia (outside of its natural range)

Eastern redcedar (Juniperus virginiana) has been expanding its range but it has also been planted all over the country (When Are Native Species Inappropriate for Conservation Plantings? Amy C Ganguli, David M Engle, Paul M Mayer, Samuel D Fuhlendorf Rangelands: December 2008, Vol. 30, No. 6, pp. 27-32. ).

European wild rabbit (Oryctolagus cuniculus), which is native from the Iberian Peninsula, southern Europe, and was naturalized in many European regions. It was later introduced to many other areas worldwide

I wonder what fraction of these are considered "invasive" versus simply "expanding their range".

House finches are native to western North America but have been naturalized all over eastern and central North America.  

You  want to double check these, but there are 2 (+1)possible plant examples that come to mind depending on how you define a continent.

- Dysphania ambrosioides (Formerly Chenopodium ambrosoides- Wormseed, Epazone or Paico), which is supposeldy native as far north as Southern Mexic, but I know occurs in California, even in Davis (it is considered an introduced weed in CA), and probably in other western states. You'd have to check the degree of naturalization. It might just be associated to agricultural areas.

- Eichornia crassipes- this is supposedly native to the Amazon basin, but it appears to have been introduced and done well in other parts of Northern South America and Central America (see the following link: http://www.cabi.org/isc/datasheet/20544)

-Varieties of Yarrow , this is just something I remember vaguely and you might have more information on, but its possible that there are some varieties of western yarrow endemic to different states that are now found in others. I am not sure about the naturalization part.

brown headed cowbirds I believe fall into this category-I read that they spread with humans altering habitats in their favor

How about coyotes in the Eastern half of NA?

Some that occur to me are black and brown rat all over Eurasia, opossum in western North America, koala in much of range in South Australia, raccoon in Alaska, muntjac in France, European rabbit in various parts of Europe, Eastern gray squirrel and fox squirrel in western US, various large African mammals (blesbok, white rhino, etc.) into protected areas and game reserves outside natural distribution.

You may want to check the DAISIE database (http://www.europe-aliens.org) where you find species native to some areas of Europe but alien (and invasive) to others. For example Arion vulgaris, http://www.europe-aliens.org/speciesFactsheet.do?speciesId=52937#, check link distribution

http://www.europe-aliens.org

Here is one bug example. The western conifer seed bug (Leptoglossus occidentalis Heid.) is a species in the family Coreidae (Hemiptera). Its native range includes the temperate and warmer regions of the Pacific coast of North America (California, Oregon and Nevada) where it can be quite common. In recent times this bug has been expanding its range eastward (on its native continent) and has been located as far east as Pennsylvania and Nova Scotia. Today, its range extends across the northern United States into Canada. More recently it has also become an invasive species in parts of Europe (first record from Italy in 1999) where it occurs as far east as the Balkans.

Barred owls

Look into the situation of Pacific salmon, Oncorhyncus spp., being intentionally introduced into the Great Lakes of eastern North America.  These have also been introduced elsewhere intentionally as a game fish, and they have maintained reproductive populations in the new regions for decades.

Schizolobium parahyba is a tree from the coastal Atlantic forests of São Paulo state, Brazil. When artificially introduced to the western part of São Paulo, where savannas and seasonal forests predominate, Schizolobium parahyba behaved as invasive. So we have a curious case of a plant native to a state and invasive in the very same state.

Abreu, R.C.R. De, Santos, F.F.D.M., Durigan, G., 2013. Changes in plant community of Seasonally Semideciduous Forest after invasion by Schizolobium parahyba at southeastern Brazil. Acta Oecologica 1–8.

The Mitred Parakeet (Psittacara mitrata), normally is distributed in lowlands and subtropical andean woods from Peru to Argentina. Few individuals were accidentally released in La Paz city and right now there are a naturalized population that exploit resources inside the city.

Lots of european examples

House Sparrow Passer domesticus; The "murder slug" Arion vulgaris, the garden snail Cornu aspersum, many other slugs and snails. Not sure what Content is in the context of Eurasia, but also Collared Dove. I am sure there are many more

There are known cases of  marine molluscs native from the European Mediterranean coasts that have become naturalized in the Atlantic coasts of Europe. For example Cyclope neritea and Hexaplex trunculus. They have been supposedly introduced as accompaying species of aquaculture target species and have built stable reproductive populations in the Bay of Biscay.  The Canadian oyster (Crassostrea virginica) from the East coast of North America has been introduced to the West coast of North America and there it has formed stable reproductive populations. Mytilus trossulus native from the European cold region have been found further South in the Bay of Biscay.

Cattle Egret has spread from Africa to several continents and Great-tailed Grackle spread from Gulf Coast to California and other states. 

India is only a subcontinent, but we have had the Red Pierrot butterfly (Talicada nyseus) extend its distribution from peninsular India to the Himalaya; besides, the Tawny Coster (Acraea violae) has extended its distribution from India to Singapore; The Bath White (Pontia daplidice) has extended its distribution from the northern face of the Himalaya to the southern face and has proceeded to the hills of NE India. The N. American Dryas julia has naturalized itself in Thailand and Singapore. If you would like references for these, I would be happy to send.

Animals: lots of fishes are spread outside their natural range. In Europe Abramis brama, Rutilus rutilus, Scardinius erythrophthalmus, Carassius carassius, Perca fluviatilis, Phoxinus phoxinus, Thymallus thymallus, Barbus barbus etc. The opening of the Danube-Rhine canal has resulted in the invasion of lots of Ponto-Caspian species, especially crustaceans e.g. Dikerogammarus villosus, D. haemobaphes, Jaera istri, Obesogammarus obesus. It's interesting to note that both taxon groups don't have a terrestrial dispersal mechanism.

Plants: can't think of too many examples although in Britain Nymphoides peltata and Stratiotes aloides are two native species with a restricted natural distribution that regularly appear as garden escapes. Presumably this is because many aquatic plants have reasonably good dispersal mechanisms at a continental scale.

yes, the barred owl in the us Strix varia, and jackals in Europe Canis aureus

In continental Chile, the Lycalopex griseus is a native fox. However, in Tierra del Fuego Island there were exotic, and introduced during the 60's to control rabbits. Now, there are a plague there, and authorities developed a local hunting licence. Also they are now threatening a local native subspecies of Culpeo fox (L. culpaeus).

A couple of plant examples within a single country, the UK:

Beech (Fagus sylvatica) is native in the south of England and Wales, but is widely planted elsewhere and is considered damagingly invasive further north in England and in Scotland. This is an example of a species which would probably have naturally extended its range further north in historical times if it had not been for woodland fragmentation due to human activity and to climate change.  The precise balance between the two factors is much debated and not relevant to the current discussion.  However, the main reason why it is considered damagingly invasive is its ecological difference from native tree species.  Beech is a 'high forest' species, regenerating in the shade and casting a deep shade - unlike any native Scottish tree.  Most native Scottish trees are relatively poor at regenerating under a canopy, and those which do are either not dominants (e.g. holly) or don't cast such a deep shade themselves (e.g. ash).  As a result, the native Scottish ground flora and epiphytes tend to be excluded from stands where beech becomes dominant, leading to an impoverished flora.  The species characteristic of native beech woodland are absent from Scotland, as they were never able to spread naturally this far north - for the same reasons as beech itself.  The fungal community can be diverse but (I think) largely with relatively common widespread species and possibly some that arrived on planting material.

Sea Buckthorn (Hippophae rhamnoides) is native on the east coast of England, and possibly as far north as Dunbar in the south of Scotland.  It has been planted elsewhere, and is generally regarded as a conservation problem as it tends to invade other dune habitats and change the nutrient status of the soil where it grows.  Within Europe as a whole, dunes with Hippophae rhamnoides are a conservation priority on Annex I of the Habitats Directive (see http://jncc.defra.gov.uk/pdf/Article17/FCS2007-H2160-audit-Final.pdf for info on its current status in UK), so it has been very important to determine the precise status of this species in Britain.

There are several examples of tree and shrub species native to one region of the US naturalizing in other regions.  Robinia pseudoacacia had a narrow distribution before it was spread widely for use as a living fence.  Similarly, osage orange, Maclura pomifera, grew in only a few southern states before being planted widely, http://www.na.fs.fed.us/pubs/silvics_manual/volume_2/maclura/pomifera.htm. Juniperus virginiana expanded its range into the Great Plains after being widely planted.  Mahonia aquifolium is native to the Pacific Northwest but has naturalized in the east from ornamental plantings.  Catalpa speciosa has widely naturalized across the southeastern US after use as an ornamental tree.  

http://www.na.fs.fed.us/pubs/silvics_manual/volume_2/maclura/pomifera.htm

A majority of the freshwater fish species in the Pacific Northwest would fall into this category, as few truly freshwater species were spared by the glaciers that scoured the area during the last ice age. Largemouth and smallmouth bass, crappie, bluegills, pumpkinseeds, walleye and others are all introduced from the eastern U.S. Other naturalized freshwater species include the crayfish Procambarus clarki and red-eared sliders.

Queensland d fruit fly was native to Queensland , Australia. It was distributed by early colonial traders and is now the major fruit fly pest on the entire eAst coast. There is a paper Dominiak and Daniels 2012 I think which covers this to some extent. Human assisted modern transport

Are the main vectors. Cheers. Bernie

Many examples of this with freshwater fish. In fact fish tend to be invaders more often within their home continent (but novel watersheds) than farther afield. Check Moyle and Ruesink as some authors for references.

High mountain lakes are naturally fishless. However, many species of trout and also minnows are being introduced, from home or from different continent.

Article Evidence of exotic trout mediated minnow invasion in Pyrenea...

Article Historical use, fishing management and lake characteristics ...

Lythrum salicaria (purple loosestrife) - native to Europe, Asia, northwest Africa, and southeastern Australia but very aggressive species for example to North America

Acer platanoides (sycamore maple) - native in Europe but invasive in North America.

These only two examples species plant but we have more :)