As a solo game designer, I can buy assets for games. But I've concluded that animations for everything would be too difficult to find or too expensive. So I'd like to think of some game types that don't necessarily need animation, or only require the most basic of animations (maybe an explosion or particle effect here or there). 

The main idea that comes to mind is a collectible card game. Or I could literally go text-based rpg game. Any other ideas?

Certain Visual Novel titles have no animations, just still images that shift.

Many turn-based or event-based games don't require them.  Animation can mean simple motion and does not necessarily mean fancy graphics, there were (and are still) many active games that have been implemented in text. 

There are text-based implementations of many long-lived games. Tetris clones, Breakout clones, Pac Man clones, Space Invaders, Snake, minesweeper, various maze games, Air Traffic Control, missile defense, Frogger,  and many pong clones, all done in text fairly easily. Then there are many games that were originally text, such as the old Ski games.  On a modern text display you can do nearly anything done on the old game systems like the 2600 or Odyssey. 

Pure text games, especially those around forming words, spelling words, matching letters, or guessing text items, these can directly work without animation at all.

Need more?

Assuming you have no problem using single frames of graphics and potentially moving it around:

Scrolling shooters.
Card games (collectable or traditional)
Board games
Trading/Entrepreneur games (buy low sell high)
Strategy games.

You could do Pac-Man or a (similar) without any animations but the graphics on that are pretty simple to draw yourself you might as well animate it.

There are assorted empire builder or entrepreneur games where pretty much all you see is static graphics on a few screens and the player clicks buttons to engage in various actions.

You can make any type of game without animations, you just have to be creative.  I can make a first-person shooter without any animations if I wanted to.

You need to be more precise in what you're asking.

Yes animations are hard to do well and takes alot of time.

My go-to game genre would be turn-based strategy, they are pretty viable without it. Look at civilization 1, 2 or the management portion of total war series (it's actually 3d but you get the idea, it could have been 2d images as well). Hearts of iron, crusader kings etc. All those games are "without animations". Static images of units, buildings etc and interface graphics are used heavily though.

Hi, I'm new.

Instead of searching for which types of game require no animation, I would reverse the process.

Find the type of game you really want to make and then design the game in consequence.

Like it was said, all types of games can be made without or very little animation.

I was actually thinking about this same topic.  I'm primarily a writer with some very basic programming knowledge, and some slightly better modeling skills, but a total failure at animation. I need to put together a final project for my degree, and I decided I'm going to do a 1st person "walking simulator" (like Gone Home and Dear Esther).  I'm making my lack of animation a part of the story, taking my weakness and working it into the game on purpose. 

Basically what 0r0d and Michael Aganier said.  Pick what you want to do, and then find a creative way to work around that. 

17 minutes ago, JediEwok said:

I was actually thinking about this same topic.  I'm primarily a writer with some very basic programming knowledge, and some slightly better modeling skills, but a total failure at animation. I need to put together a final project for my degree, and I decided I'm going to do a 1st person "walking simulator" (like Gone Home and Dear Esther).  I'm making my lack of animation a part of the story, taking my weakness and working it into the game on purpose. 

Basically what 0r0d and Michael Aganier said.  Pick what you want to do, and then find a creative way to work around that. 

Often it's the limitations that result in greater creativity and freedom because you're forced to think in different ways, in new ways.  IMO what keeps people from being truly creative and exploring new and possibly interesting ideas is the over-abundance of options.  Too many choices can cause indecisiveness, and you tend to go the safe route.

I think one could take some genre and choose whatever is the most important thing in that game type and then say "what if I limit this, or what if I remove it altogether?  What if I turn it on its head?  What game can I make then?"

Spaceship / airplanes / vehicles you might be able to get away without animations, just particle effects.