Well, I know which one is the smallest

Let's compare them all to powers of $\displaystyle 5$ .

$\displaystyle 5555 < 5^6 = 15625$

$\displaystyle 555^5 < 625^5 = (5^4)^5 = 5^{20}$

$\displaystyle 5^{555} = 5^{555}$

$\displaystyle 55^{55} < 125^{55} = (5^3)^{55} = 5^{165}$

Put simply:

$\displaystyle 5555 < 5^6$

$\displaystyle 555^5 < 5^{20}$

$\displaystyle 5^{555} = 5^{555}$

$\displaystyle 55^{55} < 5^{165}$

From these observations, clearly $\boldsymbol {5^{555}}$ is the largest.

As we don't know which one is greater, first assume that the greater number is found among $5^{555}$ & $55^{55}$ .. So, let's check! $5^{555} ? 55^{55}$ .. $\rightarrow (555*\log_5{5}) ? (55*\log_5{55})$ $\rightarrow 555 > 136.94$ So, $5^{555} > 55^{55}$ ... Now check for $5^{555}$ & $555^5$ ....

$5^{555} ? 555^{5}$ $\rightarrow (555*\log_5{5}) ? (5\log_5{555}$ $\rightarrow 555> 19.63$ So, $5^{555}>555^{5}$ Now it is clear that \boxed{\color\red{5^{555}}} is the largest among the given numbers...