$52^{52}$

$52^{2}$ = 2704

$52^{3}$ = 140 608

$52^{4}$ = 7 311 616

$52^{5}$ = 380 204 032

$52^{6}$ = 19 770 609 664

.

.

$52^{8}$ = 53459728531456

As you can see, a pattern is present $52^{4 }$ and $52^{8}$ 's last digits are same therefore.

$52^{x}$ 's last digit = 6. (Where x is a multiple of 4)

Therefore $52^{52}$ (which is equal to 170676555274132171974277914691501574771358362295975962674353045737940041855191232907575296)

's last digit is $\boxed{6}$