Solution:
First we must notice that for $1 \leq a, b$ we have $a < b$, then $a + 1 / a < b + 1 / b$. This is basic to any estimation.

The obvious approach is to notice that if $a_i \leq n$, then $a_{i+1} \geq a_i + 1 / n$. Hence it takes at most $n$ steps to get from $n-1$ to $n$. Unfortunately, this does not quite work: we need $2 + 3 + \ldots + 14 = 104$ steps to get from $1$ to $14$.

The trick is to notice that $a_{n+1}^2 > a_n^2 + 2$. But $a_2 = 2$, so $a_n^2 > 2n$. That gives $a_{100}^2 > 200 > 14^2$.