Let $X$ be a $m \times n$ random matrix whose entries are i.i.d. random variables with mean $0$ and finite variance $\sigma^2$. Let $$ Y_n =\frac{1}{n} X X^T.$$ Then according to the Marchenko-Pastur theorem, as $m, n \rightarrow \infty$ and $m/n \rightarrow c \in (0,\infty)$ the largest eigenvalue is bounded by $$ \lambda_{\max} =\sigma^2 \left (1+ \sqrt{\frac{m}{n}} \right )^2. $$
Now let's assume $n$ remains fixed while $m \rightarrow \infty $. Can one formally show that the largest eigenvalue is not bounded and that its rate of increase is related to $m$?