"Infinitely Nested" Tree-Log

First, it is best to express $\large \lim \limits_{n \to \infty} a_n$ in a simpler way. Let $\large S$ denote this expression.

$\large S = \lim \limits_{n \to \infty} a_n$

The limit can be expressed as an infinitely nested logarithmic expression:

$\large S = b\log(a_1 + b\log(a_1 + b\log(a_1 + ...)...))$

...which can be expressed as:

$\large S = b\log(a_1 + S)$

Since $\large b$ is an odd positive integer, the equation can be rearranged as:

$\large a_1 + S = 10^\frac{S}{b}$

By arduous trial and error, we come up with some possible "practical" solutions:

$\large a_1 = 1, b = 9, S = 9$ $\large a_1 = 4, b = 2499, S = 9996$ $\large a_1 = 9, b = 1, S = 1$

Since $\large a_1 > 1$ , the least possible perfect square value for $\large a_1$ is $\boxed{4}$ .