Counter example of Zorn's lemma when we only take countable chains

Consider the collection of countable subsets of $\Bbb R$, ordered by inclusion.

Every countable chain has an upper bound, since the countable union of countable sets is countable; but there is no "maximal countable subset".

(It is consistent for the axiom of choice to fail and $\Bbb R$ to be the countable union of countable sets, in which case this example is not a counterexample; and the same goes for Brian's example. But if the axiom of choice fails that bad, then there are other counterexamples to your question.)

The first uncountable ordinal, $\omega_1$, is an example: every countable subset of $\omega_1$ has a least upper bound in $\omega_1$, but $\omega_1$ itself has no maximal element.

Here's an example from topology.

Given some complete topological space (like $\mathbb R$), we can consider the set of open dense subsets of $\mathbb R$. It's clear that there isn't a minimal element (order by inclusion) in this set (remove any element from an open dense subset and its still open and dense), but the Baire Category theorem implies that every countable chain has a bound.