Boss-rush games are those where you fight bosses one after another, with no breaks. An example is the game Furi, broken down into around 13 levels, each with their own separate boss.
What was the first boss-rush game?
This is not an easy question to give a precise answer to. There are plenty of games which involve overcoming a series of challenges with no breaks, but what counts as a "boss" is harder to define. In most games "boss" refers to an enemy that is tougher and more difficult than normal enemies, but a game being exclusively a "boss rush" implies there are no normal enemies to conpare to! The idea of having a "boss" enemy in each level is possibly first attributable to a 1975 game simply called dnd, which was based on Dungeons and Dragons, but "boss rush" is harder to pin down.
My personal first thought was Space Invaders: you fight against waves of enemies with no breaks, and the enemies speed up as you kill them, reaching maximum speed with whichever is the final enemy -- meaning any of the identical enemies could potentially be a "boss". However, after doing some research on your question I saw a much better suggestion: Mike Tyson's Punch-Out!! (NES, 1987), a boxing game where you fight against 12 distinct opponents all of which have different moves and fighting styles and most of which require skill to consistently beat (the first opponent, Glass Joe, is something of an exception, intended to ease players in to the game before they reach the truly challenging fights). Back when the game was popular, most people had never even seen the Tyson fight, and they'd certainly not lasted very long if they got there.
My initial draft of this answer has that as my conclusion, and provided a few examples of earlier games with at least some sort of boss rush at some point. These included the 1985 games Space Harrier (which I've seen cited as being The First Boss Rush) and the 1985 NES game Ghosts 'n' Goblins, which featured a section in act 6 where you fight 3 bosses from earlier chapters. However, I saw there were other games from 1987 that could be argued for, including Final Fantasy (all four fiends reappear in the final dungeon ahead of the true final boss) and the original Mega Man (which has a refights of all six Robot Masters before progressing to Dr. Wily, and probably popularized the trope as it pertains to 2d platformers). I saw however these were released in December 1987, so I went to look up when precisely Mike Tyson's Punch-Out!! released.
It was at this point I discovered two things: firstly, Tyson's Punch-Out for NES released in September 1987, so those options were moot. And secondly, Mike Tyson's Punch-Out!! was not the first Punch-Out game.
The original Punch-Out was released in Japanese arcades on February 17, 1984. It is certainly the answer to your question and possibly the very first example of a Boss Rush in any type of video game, coming out more than a year before any other game I can find mentioned as having one. A game that only existed because Nintendo had a stockpile of arcade screens left over after running out of arcades to sell Donkey Kong to, and they needed an idea on what to use them for.
So there you have it, I hope this was as interesting to you as it was to me!
If fighting games are considered "boss rush", then my guess will be Samurai, an old arcade game developed and released by Sega in March 1980.
The main character is a samurai who fights with several opponents in an enclosed area. In the arena are four opponents - the rest of the enemies are trying to help them, throwing shurikens and using a hook with rope. If the player defeats all enemies, the master samurai has to be fought.
There are several master samurai (the cyan guy below) that can be encountered while playing, and they have to be defeated one after another.