Although there are no studies that explicitly look at 3-level sample size that I know of, the general rule of thumb is 30 units at each level but this is quickly falling out of favor. Although more units are obviously better (from an bias standpoint but also when considering statistical power), a lot of issues that many consider to be "small sample" issues can be addressed methodologically such that estimates from models with much smaller samples can be pretty stable. Rather than worry about sample size directly, it may be more useful to consider how complicated the model you want to fit is and compare that to your sample size. 

For instance, Ferron et al. (2009) showed that a Kenward-Roger correction can provide appropriate standard errors well into the single digits for a fairly simple model and Browne and Draper (2006) were able to obtain unbiased variance components with REML estimation with only 6 units at the highest level (again for a simple model). Gelman (2006) showed that Bayesian methods can produce unbiased estimates of variance components with as few as 3 units at the highest level using a carefully considered, weakly informative prior. If you are worried that 11 is not enough at the highest level, keeping the model as simple as possible is the best way to guard against possible small sample bias (beside collecting more data which is usually not a reasonable suggestion).

I wrote a review paper in Educational Psychology Review with Laura Stapleton last year about small samples with MLM that may be helpful  if you are looking for a summary of the literature or the a list of possibly relevant references. 

 Although there are no studies that explicitly look at 3-level sample size that I know of, the general rule of thumb is 30 units at each level but this is quickly falling out of favor. Although more units are obviously better (from an bias standpoint but also when considering statistical power), a lot of issues that many consider to be "small sample" issues can be addressed methodologically such that estimates from models with much smaller samples can be pretty stable. Rather than worry about sample size directly, it may be more useful to consider how complicated the model you want to fit is and compare that to your sample size. 

For instance, Ferron et al. (2009) showed that a Kenward-Roger correction can provide appropriate standard errors well into the single digits for a fairly simple model and Browne and Draper (2006) were able to obtain unbiased variance components with REML estimation with only 6 units at the highest level (again for a simple model). Gelman (2006) showed that Bayesian methods can produce unbiased estimates of variance components with as few as 3 units at the highest level using a carefully considered, weakly informative prior. If you are worried that 11 is not enough at the highest level, keeping the model as simple as possible is the best way to guard against possible small sample bias (beside collecting more data which is usually not a reasonable suggestion).

I wrote a review paper in Educational Psychology Review with Laura Stapleton last year about small samples with MLM that may be helpful  if you are looking for a summary of the literature or the a list of possibly relevant references. 

Thanks, Daniel. The answer was quite helpful.

See if these could possibly help Jörg 

http://www.stats.ox.ac.uk/~snijders/PowerSampleSizeMultilevel.pdf

http://joophox.net/publist/methodology05.pdf

This what I have written in relation to a related question

"A lot depends on your target of inference. Taking the example of pupils (level 1) in schools (level 2) - if you want to infer about the between -school variance than you need plenty of schools. If you want to also infer about a particular school you also need plenty of pupils in  that school. This issue most obviously comes up with people in households  - you can do a multilevel analysis on this if you have plenty of households but you are unlikely to be  able to say anything about any specific household as the number of members is likely to be too low for precise inference. In a similar manner multilevel models can be used to analyze data for twins where there can only be 2 people (level 1) in the twin pair (level 2).

At the other end of the scale where you have thousands of people (at level 1) in a country (at level 2)- a simulation study has shown ( and using MCMC Bayesian methods) that the method works quite well with only a dozen or so countries even when there is a discrete outcome and you are estimating cross-level interactions

How Many Countries for Multilevel Modeling? A Comparison of Frequentist and Bayesian Approaches http://onlinelibrary.wiley.com/doi/10.1111/ajps.12001/abstract

For general orientation on this - see

http://www.bristol.ac.uk/cmm/software/mlwin/mlwin-resources.html#datadesign

You can use the following software to do the power calculations - including three-level models

Optimal Design for Longitudinal and Multilevel Research Documentation for the “Optimal Design” Software

http://www.stat.cmu.edu/~brian/463/week14/od/od-manual-20111016-v300.pdf

http://sitemaker.umich.edu/group-based/optimal_design_software

If you have more complicated designs then this MLPOWSIM is a very general software tool; it will handle complex cross calssifications and discrete outcomes as well as simpler models - it works in MLwin and R - it is free and comes with an extensive manual that talks you through the procedures - more than a guide to the software

http://www.bristol.ac.uk/cmm/software/mlpowsim/

"edited from https://www.researchgate.net/post/What_is_the_sufficient_sample_size_in_multilevel_analysis [accessed May 8, 2015].

From my perspective having 3000 level 1 units is not a particularly large sample size- modern software can deal with much larger. The imbalance of very different numbers of lower level units should not be a particular problem (MLPOWSIM allows you to experiment with imbalance) but you will not be able to say anything specific about a particular  unit that does not have many lower level units - there will be quiet a lot of shrinkage (see Jones and Bullen) and appropriately wide confidence intervals - what you should not do is drop units to achieve balance unless there is something very different about those units with  low numbers - that is you do not want to use random effects modeling to pool across them.

https://www.researchgate.net/publication/225303298_CONTEXTUAL_MODELS_OF_URBAN_HOUSE_PRICES_-_A_COMPARISON_OF_FIXED-COEFFICIENT_AND_RANDOM-COEFFICIENT_MODELS_DEVELOPED_BY_EXPANSION

Article Contextual Models of Urban House Prices: A Comparison of Fix...

As mentioned above, Optimal Design is great for this.  It can do power analysis for multilevel designs and is pretty easy to use.  You should be able to put in the numbers from your question (although the imbalance may make it more difficult) and come up with power given different sample and effect sizes.  If you don't have effect sizes, you can come up with various estimates of your expected effect sizes and it will give you the sample size needed for your desired power.  It is pretty flexible and practical. 

I too have the problem of sample size for a three level study. I too like Jorg, am at a quandary regarding how many third level units to select. I suppose 5 is far too small if quasi likelihood in MLwiN is used for estimation.  Daniel and Kelvyn I find your discussions most enlightening. Thanks.