Since the baby arrived, my brother has barely ____ more than four hours a night.

Explication

Since the baby arrived points to a stretch reaching the present, so the sentence takes the present perfect and needs a participle: slept. Sleep supplies that one past form only, and it is the same word used alone in he slept badly last night, which means the auxiliary has, not the verb, carries the perfect meaning. Sleeped copies a regular pattern this verb does not follow.