A countale partially ordered set that has an uncountable number of maximal chains
Solution 1:
Take $S$ to be the set of all finite binary strings, and say $a\lt b$ iff $a$ is a prefix of $b$. Then every infinite binary string defines a maximal chain in $S$, and there are uncountably many infinite binary strings.
You can picture this same example as an infinite binary tree, which has countably many nodes but uncountably many branches; here $a\lt b$ iff $a$ is an ancestor of $b$.