= Solution
For each one-particle state $i$, a <boson> occupation number can be any $n_i\geq0$. Its contribution to the <grand canonical partition function> is the <geometric series>
$$
\sum_{n_i=0}^{\infty}e^{-\beta n_i(\epsilon_i-\mu)}
=\frac1{1-e^{-\beta(\epsilon_i-\mu)}}.
$$
The noninteracting modes factorize, so
$$
\mathcal Z=\prod_i\frac1{1-e^{-\beta(\epsilon_i-\mu)}},
\qquad
\boxed{\log\mathcal Z=-\sum_i\log\left(1-e^{-\beta(\epsilon_i-\mu)}\right).}
$$
Convergence requires $\mu<\min_i\epsilon_i$. Since the lowest energy is zero, the limiting maximum <chemical potential> is
$$
\boxed{\mu_{\max}=0.}
$$
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