It's quite straightforward to construct a (complete) lattice in which no chain has maximum cardinality: for each $n\in \omega\setminus\{0\}$ let $C_n$ be a copy of $n$ with the chain ordering inherited from $n$. Put all the $C_n$'s side by side, and add a bottom and a top element.

However, this lattice is highly non-distributive, which leads to the following question:

Is there a distributive lattice $L$ such that for every chain $C\subseteq L$ there is a chain $C'\subseteq L$ such that $|C|<|C'|$?