Nice catch. Wikipedia and PlanetMath give definitions with this clause: "if δ is limit, then α^δ is the limit of the α^β for all β < δ." However, MathWorld separates the case of the zero base: "If is a limit ordinal, then if , . If then, is the least ordinal greater than any ordinal in the set ."
The first Wikipedia definition in terms of ordered sets says the following.
According to this, is the set of functions from to , i.e., .In general, any well ordered set B can be raised to the power of another well ordered set E, resulting in another well ordered set, the power B^E. Each element of B^E is a function from E to B such that only a finite number of elements of the domain E map to an element larger than the least element of the range B (essentially, we consider the functions with finite support).