It's quite straightforward to construct a (complete) lattice in which no [chain](https://en.wikipedia.org/wiki/Total_order#Chains) 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](https://en.wikipedia.org/wiki/Total_order#Chains) $L$ such that for every chain $C\subseteq L$ there is a chain $C'\subseteq L$ such that $|C|<|C'|$?



Chains of maximum cardinality in distributive lattices
If the following conjecture of Don Monk is true, then it yields an affirmative answer to the question.


**Conjecture.** 
If $K$ is a nonempty set of infinite cardinals, then there is a Boolean algebra $A$ such that a cardinal $\kappa$ is the size of a maximal chain in $A$ if and only if $\kappa\in K$.


(If this conjecture is true, take $K=\{\aleph\_n|\;n\in\omega\}$ and you have your example.)


This conjecture appears on the last page of


Monk, J. Donald, 
Towers and maximal chains in Boolean algebras. 
Algebra Universalis 56 (2007), no. 3-4, 337-347; [doi:10.1007/s00012-007-2002-8](https://doi.org/10.1007/s00012-007-2002-8), [author's website](http://euclid.colorado.edu/~monkd/monk77.pdf).


In this paper, Monk proves a strong partial result:


**Theorem.** (GCH)
If $K$ is a nonempty set of infinite regular cardinals, then there is a Boolean algebra $A$ with the property that a regular cardinal $\kappa$ is the size of a maximal chain in $A$ if and only if $\kappa\in K$.


Thus, Monk's Conjecture is that GCH is not needed in this theorem, and the restriction to regular cardinals is not needed. 


