To the limit of one's efforts - to the last extremity.
Bitter has been an adjective meaning acrid or sour tasting since the year 725 AD at least. The word was in common use in the Middle Ages and Shakespeare uses it numerous times in his plays and poems, as do many other dramatists. The phrase 'the bitter end' would seem, fairly obviously, to come directly from that meaning.
But not so fast. Enter, stage left, Captain Smith. Here's what he has to say, in his publication Seaman's Grammar, 1627, which is the earliest citation of the phrase in print:
"A Bitter is but the turne of a Cable about the Bits, and veare it out by little and little. And the Bitters end is that part of the Cable doth stay within boord."
As you might have deduced, a bitt is a post fastened in the deck of a ship, for fastening cables and ropes. When a rope is played out to the bitter end, it means there is no more rope to be used.
But again, not so fast. Folk etymologists are those who say something is true with no more justification than that they would like it to be true. They are thickest on the ground in the area of military and especially naval attributions. People seem to love a sailor's yarn, and anything with a whiff of the sea is seized on with enthusiasm. So much so that more thoughtful etymologists have dreamed up the inventive acronym CANOE - the Committee to Ascribe a Naval Origin to Everything.
So, is this one from CANOE or not? We like to be definitive and, although the naval origin does seem to have a good case, it isn't conclusive. This time we'll sit on the fence and let you decide.