どんなに彼をほめてもほめきれない。

how; how much

he; him; his; boyfriend

indicates direct object of action; indicates subject of causative expression; indicates an area traversed; indicates time (period) over which action takes place; indicates point of departure or separation of action; indicates object of desire, like, hate, etc.

to praise; to admire; to speak well

too; also; in addition; as well; (not) either (in a negative sentence); both A and B; A as well as B; neither A nor B (in a negative sentence); even; as much as; as many as; as far as; as long as; no less than; no fewer than; even if; even though; although; in spite of; further; more; again; another; the other