Let $\mathbb{N}$ be the set of natural numbers with the discrete topology. Is there a compactification $X$ of $\mathbb{N}$, other than the Stone–Čech compactification $\beta \mathbb{N}$, in which every infinite subset of $\mathbb{N}$ has at least two limit points?
In other words, does there exist a compact Hausdorff space $X$, not homeomorphic to $\beta \mathbb{N}$, having a countable dense discrete subset $A$ such that every infinite subset of $A$ has at least two limit points?
Motivation: $\beta \mathbb{N}$ has the property that for every infinite $B \subset \mathbb{N}$, the closure $\bar{B}$ is again homeomorphic to $\beta \mathbb{N}$ (see Lemma 5 here). In particular, $B$ has $2^\mathfrak{c}$ limit points. Of course, $\beta \mathbb{N}$ is the "largest possible" compactification of $\mathbb{N}$.
On the other hand, if $X$ is first countable at any point $x$ of $X \setminus \mathbb{N}$, then there is a sequence in $\mathbb{N}$ converging to $x$. This sequence is thus an infinite set with only one limit point. So any first countable compactification is "too small".
I am wondering what is in between.
(This came up while thinking about this answer.)
Take two points $x,y\in\beta\mathbb{N}\setminus\mathbb{N}$, and let $X$ be the quotient of $\beta\mathbb{N}$ obtained by identifying $x$ and $y$. You can see that $X$ is not homeomorphic to $\beta\mathbb{N}$, since $\beta\mathbb{N}$ is the Stone-Cech compactification of the subspace consisting of all its isolated points and $X$ is not.