As I've mentioned before, it took me around 15 years and hundreds of tries to beat the Gauntlet challenge mode in Wing Commander Academy, so no doubt the two times I've finished that are on top.
Generally driving games are most intense genre, if you count that as battle. In case of Death Rally you certainly can, so the finale on that was something.
Some button-smashing beat'em ups get intense, playing on the difficulty you can just barely beat after dozen or two tries. Darkstalkers series and Queen of Heart '98 come to mind.
In shooters and platformers, actual boss fights tend to be tiresome.
Honorable mention: Magic: The Gathering - in the long major single-player mode, once you defeat the presumed final enemy there is yet one more challenge: Arkazon. While normally in Magic players have 20 life points (and you lose when falling to zero) and in this campaign mode player and opponents may have 30 or so, this final opponent starts with 400 points! (I think, might've been 300 or 500?)
In Magic you must have deck of minimum of 40 cards and usually it's tactically smarter to have as small deck as possibly. However with opponent with 20 times normal life, it'll be very very long game, if you can survive long yourself, and you lose if you run out of cards. Fortunately I have a habit of using stupidly huge decks, so I went to this - without knowing what was coming - with 150 or 200 cards. The opponent has game's maximum 499 card deck of all colors, which makes it really bad deck, but with those life points that doesn't really matter.
And somehow I managed to win that. Playing with top difficulty. (Because the damn game has only one difficulty setting, so if you want to play a shorter game, it also makes your opponent so stupid they are no challenge at all.) First try. (And I wouldn't have tried again because getting to end from last save point was an hour or at least half of work.) And I recall I was left with less than 20 cards in the end, I was that close to losing (having in the beginning also fallen momentarily to under 10 life points).
So, for a card game, that was pretty intense.