Used to indicate an upper limit or the most optimistic estimate. Often paired with 少なくとも (at least) for contrast.
See also: 少なくとも
This work will be finished in three days at most.
It's a small venue that can hold at most about a hundred people.
多くとも means 'at most', setting an upper bound, the direct opposite of 少なくとも's lower bound.
Compound of 多く (many, much) + とも (even if). Literally 'even if many', it developed into a set phrase meaning 'at most'.