Let $R = \mathbb{F}_2^{\mathbb{N}}$ be the countable product of the two-element field with pointwise addition and multiplication. I know that every prime ideal in $R$ is maximal (since it is a Boolean ring), and the ideals of the form $\mathfrak{m}_n := \left\{ x \in R: x_n = 0 \right\}$ are prime and thus maximal. However (if I'm not mistaken), there is another proper ideal which is not contained in any ideal of the form $\mathfrak{m}_n$.
In particular, the set $\mathfrak{n}$ of elements of $R$ with finite support (i.e. the set of all $x \in R$ such that $x_i = 0$ for all but finitely many $i\in\mathbb{N}$) is a proper ideal. For every $n$, the element of $R$ with a single 1 in the $n$th entry and zeroes elsewhere is in $\mathfrak{n}$ but not in $\mathfrak{m}_n$, so it follows that $\mathfrak{n} \not\subseteq \mathfrak{m}_n$ for all $n$. Hence $\mathfrak{n}$ is contained in a maximal ideal which is not of the form $\mathfrak{m}_n$ for any $n\in\mathbb{N}$.
However, $\mathfrak{n}$ is not maximal because it is not prime. Indeed, if $x = (1,0,1,0,1,0,\ldots)$ and $y = \mathbf{1} - x$, then $xy = \mathbf{0} \in \mathfrak{n}$ but neither $x$ nor $y$ is in $\mathfrak{n}$. (More generally, if $\mathfrak{p}$ is any prime ideal in $R$, then for every $x \in R$, exactly one of $x$ and $\mathbf{1}-x$ is in $\mathfrak{p}$.)
My question: what is the maximal ideal containing $\mathfrak{n}$? (Or rather, what does this ideal look like?) I haven't been able to come up with a nice description of it. Are there other maximal ideals than those I have mentioned?
My question has been answered by user10354138 and Qiaochu Yuan in the comments. My thanks to them! I am paraphrasing their answers here so that I can mark this question as answered.
In short, there is no "nice" description of these maximal ideals.
The maximal ideals containing $\mathfrak{n}$ correspond bijectively to the free ultrafilters on $\mathbb{N}$. The existence of these ultrafilters follows from the axiom of choice, but it is consistent with ZF that they do not exist, so they do not have nice descriptions.