  By studying the variety of J\'{o}nsson-Tarski algebras, we demonstrate two
obstacles to the existence of large J\'{o}nsson algebras in certain varieties.
First, if an algebra $J$ in a language $L$ has cardinality greater than $|L|^+$
and a distributive subalgebra lattice, then it must have a proper subalgebra of
size $|J|$. Second, if an algebra $J$ in a language $L$ satisfies
$\text{cf}(|J|) > 2^{|L|^+}$ and lies in a residually small variety, then it
again must have a proper subalgebra of size $|J|$. We apply the first result to
show that J\'{o}nsson algebras in the variety of J\'{o}nsson-Tarski algebras
cannot have cardinality greater than $\aleph_1$. We also construct
$2^{\aleph_1}$ many pairwise nonisomorphic J\'{o}nsson algebras in this
variety, thus proving that for some varieties the maximum possible number of
J\'{o}nsson algebras can be achieved.
