Max clique in interval graph
Interval graphs are chordal graphs, so you can use a linear time algorithm for finding max clique in a chordal graph. This can be done by first finding a perfect elimination ordering, and then using the fact that every time a vertex is eliminated it forms a clique together with its neighbors that are not eliminated yet. Furthermore all maximal cliques appear in this process. 

To find the perfect elimination ordering, maximum cardinality search (MCS) is the simplest algorithm and it works in linear time. (I don't know why Wikipedia mentions lexbfs first, it is much more complicated.)