The hyperexponentiation of a number

The hyperexponentiation or tetration of a number $a$ by a positive integer $b$, denoted by $a\uparrow \uparrow b$ or $^ba$, is recursively defined by:

$$ a\uparrow \uparrow 1 = a,\ a\uparrow \uparrow(k+1) = a^{a\uparrow \uparrow k} $$

Thus we have e.g. $3\uparrow \uparrow 2 = 3^3 = 27$, hence $3\uparrow \uparrow 3 = 327 = 7625597484987$ and $3\uparrow \uparrow 4$ is roughly $10^{3.6383346400240996*10^{12}}$.

Find the last $8$ digits of $1777\uparrow \uparrow 1855$.