Could you give me a ride to the hospital?

Literal

Hospital [up-to-まで] riding letting [won't-you-くれませんか].

乗せていってくれませんか combines 乗せる ('to give a ride') + ていく ('going') + てくれる ('do for me') + ませんか (polite request). A layered benefactive request — 'won't you do me the favor of taking me to the hospital?'