It is clear that $ f_1(k) \leq (9(\log_{10}k+1))^2$ and, in particular, $ f_1(k)<k$ for all $ k\geq 400$. It is easy to find all limit cycles of $ f_1(k)$ - there are three of them: $ 1,1,1,\dots$ $ 81,81,81,\dots$ $ 169,256,169,\dots$ It is clear that $ f_{1991}(2^{1990})$ ends up in one of these numbers. Since $ f_1(k)\equiv k^2\pmod{9},$ we have \[ f_{1991}(2^{1990})\equiv 2^{1990\cdot 2^{1991}}\equiv 4\pmod{9},\] implying that $ f_{1991}(2^{1990})=256.$

what is the maximum valu of 5sinx+4cosx