2. More generally, if we define the hyper power ⁿa by

¹a:=a, ⁿ⁺¹a:=a^(na) for n>=1

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

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

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

1.  The last digit will be 3.
7ⁿ, such that n is a positive integer ending in the digit 3, where the 3 is preceded by an even digit, will end with the three digits 343 [if preceded by an odd digit, the last three digits would be 407]. Because 7⁷ = 823543, which ends in a 3 preceded by the digit 4 [an even digit], 7^(7^(7^(7^(7^(7^7))))) will end in the digits 343.