Take a sequence $X_1,\ldots,X_n$ where each $X_i\sim\mathcal{N}(\mu,\sigma^2)$ is an i.i.d. normal random variable. Denote by $X_\max$ the maximum of this sequence. A well-known fact about convergence of $X_\max$ is that $\frac{X_\max-\mu}{\sigma\sqrt{2\log n}}\rightarrow 1$ almost surely (for instance, see Example 4.4.1 in Galambos "The Asymptotic Theory of Extreme Order Statistics").
Now suppose you scale an arbitrary element $X_j$ of the sequence above by a small amount to get a new sequence: $X_1,\ldots,(1+\epsilon)X_j,\ldots,X_n$ where $\epsilon=o\left(\sqrt{\frac{\log n}{n}}\right)$. Is there a convergence result for the maximum of this almost-identically distributed sequence of independent normal random variables? It seems to that that almost-sure convergence similar to above should exist, but I am having hard time deriving it...