Circus Empire

The Gladiators: Galactic Circus Games :)

3. Painkiller has the Looney Park Level.
4. The Park seems to play in an amusement park, though not in a real circus.

Seven Games of the Soul

really creepy circus point and click

Dropsy

One of the worlds in Lemmings 2: The Tribes was circus-themed. Video

Also there are a lot of hidden object games that are circus themed, if you like these.

We counting non-PC games? If so, I occasionally see this shooter called CarnEvil at arcades around here.

I was trying to recall some adventure game where I swear I had to use a whip on a lion or something, when a thought popped into my head. Why are we trying to list circus games?

I remember I played a touch typing edutainment game on the Amiga, which I think had a circus theme. (A juggler threw balls with letters from one hand to the other, and you had to type it in time.)

And then there's the circus pinball table from 1977, which has a pretty good pc version:
http://store.steampowered.com/app/623790/Zaccaria_Pinball__Circus_2017_Table/

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Sanitarium: Had a weird circus in the middle of a swamp
Painkiller: Battle out of Hell and Overdose: Looney Park level
Blood: Episode 1 Level 4 was called Carnival of the Dead IIRC
Baldur's Gate 2: The area right after you exit the first dungeon
Lego Batman 2: It was on the North Island
Carmageddon 2: Had a course set in a circus/carnival

There was an amusment park in STALKER: SOC but not sure if that counts

If I'm not mistaken it's the level with the Duke Nukem easter egg, right?

I don't know if it counts Fairfox, but Max Payne 2 has a funhouse level. You actually go there a couple of times, but only once it plays out like an actual walk through the funhouse.

Yup, Duke is in the area with Jojo's tight rope walk.

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First and only game that comes to mind for me personally is Circus Charlie on the MSX