Mechanical games have been around for quite a while. The predecessor to pinball, for instance, was around in the 1700s, though bumpers weren't added until the 1930s, and flippers until 1947. Still, the electronics in a pinball machine would be easy to replace with fluid, and the scoreboard could easily be lit with lights.
Similarly, the venerable Pong could be made into a purely mechanical game today, even going as far as having a wind-up motor to run the game; a fluid-driven motor would be quite simple.
Of course, you may argue that those games aren't, strictly speaking, video games, which is true. However, with just a little bit of stage-magic, it's easy to convert a purely mechanical game into a "holographic" game, through the use of curved, mirrored glass, such as that used by the game Time traveler. The game itself would, technically, be purely mechanical, but miniaturized. A bright light, shining onto the game, would cause the image to reflect from the curved screen, giving the illusion of both size and depth.
With a little mechanical work with mirrors, it wouldn't be hard to convert a tilting maze into a joystick-controlled game; pushing the joystick right would tilt the maze, as well as a series of mirrors, which would lead the player to believe that the joystick motions move their character, rather than the board itself.
Shadows could make excellent "video" displays; a light, shining past a set of figures on thin wires, would cast dark shadows. As the characters move, the wires are twisted, causing the characters to either change shape (by having different profiles), or appear and vanish (by simply being flat). A game like Space Invaders would be fairly simple to design.
One step further, you could use fully colorized "sprites", drawn on sheets of clear glass (plastic would be easier, if possible), in front of a scrolling, fully colorized background. A second, synchronized roll would have a cutout area for the player to move through. As the background scrolls top-to-bottom, the player's character can move left and right, controlled with a joystick. A third layer of sprites would contain a sheet with the 'bad guys' for the player to avoid - oncoming vehicles, falling rocks, or pitfalls in a race track. In fact, this kind of game could be made into a multiplayer game quite easily, just by adding more joysticks and player scrolls. The entire game could be purely mechanical.
On the other hand, you could make a choose-your-own-adventure style game, with full-scale video and few (or at least recycled) choices, such as FMV games. As long as you have several reels of full video tape with a system able to mechanically switch between them, it becomes almost trivially easy to write a game.
You could combine any number of these, as well; a holographic side-scrolling game, a shadow board with full-video cutscenes, or even a mechanical game with full video projected on it.
In all these cases, each machine would be nearly the same size as a modern arcade machine. In fact, using scrolling paper or video would mean that a single basic machine could have several "adventures", each with their own set of scrolling paper, meaning the manufacturer could spend more money on producing games, and less on producing game cabinets. If taken to the logical extreme, a gaming cabinet could be purchased much like as a game console is today, with replaceable cartridges to play different games, even replaceable controllers!
As long as full video type screens exist, any number of games are possible. Look through older video games, like Pac-Man, Asteroids, Space-Invaders, and so on; the amount of code to write those games is tiny, easily enough for an optically driven machine. As long as the screen exists, the game can be written for it. A hydraulic system that opens and closes tiny holes, allowing a bright light to shine through, would be a fairly simple way to implement a projection screen capable of full video; from there, it's easy to make a game, and cheaper than using optics, too.