Let $X_1,\dots,X_n \sim \mathcal N(0,I_p)$ be i.i.d., and suppose
\begin{align*}
\frac{p}{n}\to\gamma\in(0,\infty).
\end{align*}
Then the largest eigenvalue of $\widehat\Sigma$ satisfies
\begin{align*}
\lambda_1(\widehat\Sigma) \xrightarrow{a.s.} (1+\sqrt\gamma)^2.
\end{align*}