The hyperexponentiation of a number (noch nicht übersetzt)

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

a↑↑1 = a,
a↑↑(k+1) = a^(a↑↑k).

Thus we have e.g. 3↑↑2 = 3³ = 27, hence 3↑↑3 = 3²⁷ = 7625597484987 and 3↑↑4 is roughly 10^{3.6383346400240996*10^12}.

Find the last 8 digits of 1777↑↑1855.