People often think that a field has been 'picked clean' and conclude that either no further progress is possible or that further discoveries are going to be rare and incremental.

e.g. "the future truths of physical science are to be looked for in the sixth place of decimals" -- Michelson, 1894

But there are always deep problems and this implies that unlimited progress is possible. The really big discoveries can happen at any time and are of an unpredictable nature, not least because they affect multiple fields. They leave behind plenty of smaller problems for everyone to pick up.