Does anyone know of any interesting examples of an infinite abelian $p$-group which is uncountable?

By non-interesting here I mean the direct sums of cylic and quasi-cylic groups, and totally projective groups. (The class of totally projective $p$-groups is the smallest class of abelian groups which contains the cyclic group of order $p$, is closed under taking direct sums and summands, and contains a group A if and only if it contains, for every ordinal $x$, both $p^{x}A$ and $A/p^{x}A$. It contains all countable abelian $p$-groups, and all groups in this class can be described by a recursive construction.)

The only example outside of these I know is the torsion subgroup of an unrestricted Cartesian product of cylic groups. Does anyone know of any others?

2more comments