The question sounds silly, but I just read about Restriction Enzyme Star Activity. What about other enzymes which are not restriction enzymes, are they also affected by conditions that cause star activity in Restriction enzymes?
From the Promega website: http://www.promega.co.uk/resources/pubhub/enotes/what-is-restriction-enzyme-star-activity/
All restriction enzymes bind DNA nonspecifically, under optimal conditions the difference in cleavage rates at the cognate site and the next best site (single-base substitution) is very high. Under non optimal conditions eg changes of pH, type of ions present, ionic strength, metal cofactors other than Mg2+, high enzyme:DNA ratios and the presence of volume excluders (glycerol, ethylene glycol, etc.) a loss of fidelity can occur. This leads to cleavage activity at non-cognate sites.
An example of a non-restriction enzyme that has altered function as a consequence of a change in metal ion cofactors is Taq polymerase where the normal DNA polymerase activity can become a reverse transcriptase activity in the presence of Mn ions instead of Mg ions. I am not sure though that this would be referred to as "star activity"