Power Doesn't Change Last Digit?

Cases mod 10.

If gcd $(a,10) = 1$ , then $a$ is odd so $a^a \equiv a$ mod $4$ and hence $a^{a^a} \equiv a^a$ mod $10$ by Euler's Theorem.

If $a \equiv 0$ mod $10$ then both sides are $0$ mod 10. If $a \equiv 5$ then both sides are $5$ mod $10$ .

If $a \equiv 4$ or $6$ mod $10$ then both sides are $6$ mod $10$ .

So we're left with $a \equiv \pm 2$ mod $10$ . If $4|a$ then both sides are $6$ mod $10$ . So we're left with $a \equiv 2, 18$ mod $20$ . For these, the left side is $6$ and the right side is $4$ mod $10$ .

Putting it all together, we have that $a^{a^a}$ and $a^a$ end with the same digit if and only if $a \not\equiv 2,18$ mod $20$ . There are $\fbox{1814}$ such $a$ between $1$ and $2015$ .