The article "Groups with every proper quotient finite" from 1971 claims in Theorem 4 that if $\mathcal{B}$ is a Boolean algebra satisfying the maximal condition, then it must be finite.
Def: A Boolean algebra has the maximal condition if every increasing sequence $x_i$ stabilizes, namely, there exists $i_0 \geq 0$ such that $x_i = x_{i_0}$ for all $i \geq i_0$.
Any idea how this can be proved?
Comments: By the Stone representation theorem, you may assume that the elements are clopen subsets of a compact Hausdorff totally disconnected topological space.
For convenience I will construct a decreasing sequence (and then you can of course take complements to get an increasing sequence). Say an element $x\in\mathcal{B}$ is infinite if there are infinitely many $y\in\mathcal{B}$ such that $y\leq x$. Note that if $x=a\vee b$, then at least one of $a$ and $b$ is infinite, since every $y\leq x$ is the join of an element below $a$ and an element below $b$ (namely $y\wedge a$ and $y\wedge b$). So, if $x$ is infinite, then there exists $y<x$ that is also infinite (just take $a$ such that $0<a<x$ and then at least one of $a$ and $x\wedge \neg a$ must be infinite). Starting from $1$ which is infinite by hypothesis, you can then repeat this process to construct a strictly decreasing sequence of infinite elements of $\mathcal{B}$.