It depends which "0" you are talking about. If "0" is the Natural Number then 0:={} and 0^0:=#{all functions from 0 to 0}.
As 0={}, there is only one map from 0 to itself, ie. {} (ie. the function with dom = {} and range = {}). So in this case the answer is 0^0 = #{{}}=1 (because we want the size of the set of functions and there is only 1 function).
However, if we get to the real numbers, 0 is not defined in this way. In fact it's definition is not important. What is important is that 0^0:=exp(0 ln 0) but as ln 0 is not defined, 0^0 is, by definition, not defined. So 0^0 when 0 is a real number has no definition. (This definition stands up because exp x is not simply e^x; exp x is the series by definition.)
Confusion arises when people start talking about taking the limits of stuff like x/x as x->0. However, a limit of a function at a certain point is not necessarily the value of the function at that point. In fact for this to be true it is necessary and sufficient that the function be continuous. And by strict definition, if you cannot simply calculate the value of a function as it is written then it has no value, even if it may have a limit. For example x/x has no value at x=0 because 0/0 is not defined (this is because x/y is defined as the unique number z such that zy =x and it is obvious that no such number exists for x=y=0) whereas x/x as x->0 has limit 1.