I think you need more than (just) the last digit of 7^7.

Notice that 7 = 7 (mod 10)

7^2 = 49 = -1 (mod 10)

7^3 = 7*-1 (mod 10) = -7 (mod 10)

7^4 = 1 (mod 10)

7^5 = 7 (mod 10)... now we have repeated.

So you need to know what 7^7 is (mod 4), since the pattern is length 4. This will require the last TWO digits of 7^7, which are 43 = 3 mod 4. So the last digit of 7^(7^7) should be -7 = 3.

I've tried to color-code a little so you can see the train of thought