Nested Fractional Exponent Discovered

The sequence $(-1), {\large (-1)^ \frac{1}{(-1)} }, {\Large (-1) ^ \frac{1}{(-1)^ \frac{1}{(-1)}} }, \ldots$ clearly converges to the integer $-1.$

The sequence $(4), {\large (4)^ \frac{1}{(4)} }, {\Large (4) ^ \frac{1}{(4)^ \frac{1}{(4)}} }, \ldots$ converges to the integer 2. (Can you prove it?)

Does there exist another value $b \neq 1, -1, 4$ such that the sequence $\{a_n\}$ defined recursively by $a_0 = b; \quad a_{n+1} = {\large b^{\frac{1}{a_n}} }$ also converges to an integer?

Note: Written out, this sequence is $(b), {\large (b)^ \frac{1}{(b)} }, {\Large (b) ^ \frac{1}{(b)^ \frac{1}{(b)}} }, \ldots.$