^{777777}= 7^(7^(7^(7^(7^(7^7))))) ?

2. More generally, if we define the hyper power ^{n}a by

^{1}a:=a, ^{n+1}a:=a^{(na)} for n>=1

does the series of the last digits of ^{1}a, ^{2}a, ^{3}a,... always become periodic at some point?

If so, can you provide a sharp upper limit for the period length?

Note: The hyper power ^{n}a is also often denoted as a^^n.