profusely

Profusely means in such abundance that it pours out unchecked. Sweat, blood, thanks or apologies.

It attaches to a small and oddly specific set of verbs, since bleeding profusely, sweating profusely, apologising profusely and thanking somebody profusely account for nearly every use of the word. The first two are physical and the last two are social, and all four involve more than would be expected. The adjective profuse is far rarer, and profusion means an abundance.