This problem is an example of order statistics with non-identical distributions; in this instance, due to having non-identical parameters. We are given a $N(\mu, \sigma^2)$ parent where identicality is relaxed by replacing $(\mu, \sigma)$ with $(\mu_i, \sigma_i)$ for $i = 1, ..., n$, with parent pdf say $f_i(x)$:

and cdf $F_i(x)$.
The general form for the pdf of $Z = \max \left(X_1, X_2, \dots, X_n \right)$ will be:
$$\text{pdf}(Z) \quad = \quad \sum _{i=1}^n \big\{ \; f_i(z) \prod _{j=1}^n F_j(z){}_{\text{for } j\neq i} \big \}$$
This is manageable. To illustrate and check, consider a specific case of say $n = 4$. Then the pdf of the maximum is the pdf of the $4^{\text{th}}$ order statistic in a sample of size 4, which can be evaluated automatically using the OrderStatNonIdentical
function in the mathStatica package for Mathematica:

where the Normal cdf $F_i(x) = \frac{1}{2} \left(1+ \text{Erf}\left(\frac{x-\mu _i}{\sqrt{2} \sigma _i}\right)\right)$, and Erf
denotes the error function.
This can yield unusual functional forms, such as bimodal pdfs, by choosing suitably designed parameters. For example, here is a plot of the pdf when $n= 4$, and $\mu _1\to 4,\mu _2\to 3,\mu _3\to 2,\mu _4\to 7$ and $\sigma _1\to 0.1,\sigma _2\to 2,\sigma _3\to 3,\sigma _4\to 4$:

More usual would just be a skewed right pdf.
Notes
1. As disclosure, I should add that I am one of the authors of the function used above.