First of all, I'm pretty sure $a_{n+1}$ is supposed to be $\frac{a_n}2$ when $a_n$ is even, not $n$ (and $a_n+7$ when $a_n$ is odd). Second of all, I think $a_1=1993^{1994^{1995}}$, that is, the topmost number is $1995$, not $1994$. Every number $>7$ is turned into one $<7$ in at most $2$ steps, so the minimum is $\le 7$. Since our initial term is not divisible by $7$ and it's clear that the rules can't produce terms divisible by $7$ when there are none, it means that the minimum is $\le 6$. $6,5,3$ go to $3$, while $4,2,1$ go to $1$, so there are two possibilities: the minimum is either $3$ or $1$. $2$ is a quadratic residue modulo $7$, so either all the terms of the sequence are quadratic residues modulo $7$, or none are. Since the initial term is $4\pmod 7$, it means that all terms are quadratic residues of $7$, so the minimum can't be $3$, meaning that it must be $1$.