Can an aqueous solution conduct electricity forever?
Since nobody has posted a counter example, I will go ahead and say that my claim is true for direct current (which is what I originally had in mind). The current drives a chemical reaction and, at some point, the reactants will run out and the solution will no longer have enough ions to effectively conduct electricity. 

Hopefully someone will post a counter-example and make things more interesting :)

In the case of alternating current, it seems possible to plate material onto one electrode and remove it from the other. When the current switches direction, this process would reverse and the electrodes would remain the same mass. In theory, this could go forever, but in reality I wonder if the electrodes would deteriorate and break due to the many cycles of deposition and erosion.