This question is inspired by a Google Code Jam problem (2009 1A Multi-base happiness) where it is necessary to assume there is such an integer for the set of bases $\{2, 3, \ldots, 10\}$.
Define the function $f_b:\mathbb{N} \mapsto \mathbb{N}$ such that $f_b(n)$ is the sum of the squares of the base $b$ digits of $n$.
A positive integer $n$ is "happy in base $b$" if repeated application of $f_b$ will result in a $1$.
Given a set of bases $\{b_1, \ldots, b_k\}$ (with all $b_i > 1$) is it possible to find a single positive integer $n \ne 1$ such that $n$ is happy in $b_i$ for all $1 \le i \le k$.
(edited to make it more clear the same integer has to be happy in every base)