A friend of mine has mentioned some games. I confirmed some of them but not all.
In the Mario Kart games from DS through Tour, the Blooper item will ink its targets.
In Paper Mario Color Splash, you can hit Peach with the paint hammer and cover her.
Mario Party 10 has a paintball battle mini game, where characters can be covered in blue, green, pink, and yellow paint. The coverage is great, but the only way to get a good view is if you somehow could hack the game and manipulate the camera. You can also get a Blooper to ink a character when you land on a certain Happening Space in Whimsical Waters, but that's insanely hard and time-consuming to do...plus it might only be possible in the Bowser Party mode.
In Mario Super Sluggers, the Blooper's special move can ink characters.
Apparently, in Mortal Kombat 9, you can slime characters using Reptile but it might not stay on the victim.
In the indie game A Hat in Time, during the third level of the first act, you can find mud puddles that, upon entering one, will coat your character in mud.
In Okami (and possibly the sequel), you can cover NPCs in ink if you mess up a brush stoke over them.
In Metal Gear Solid 3, when you have to escort an injured Eva, you can knock her down and have her lie face first in some mud puddles.
Bushido Blade on PS1 has a mud pit you can go into, but it's part of a huge map so finding it may take a while.
Darkstalkers 3 has Q-Bee, who can hit characters with a giant ball of honey.
Dragon's Dogma has a "tarring" status, which covers the character in oil.
6/14/22, 5:38pm: This post won't affect thread last post date.
1989's "Bar Games" includes a round called "Wet n Wild", where you play the assistant at a wet t-shirt contest, running up an down a catwalk, trying to soak a model (male or female, your choice) who won't stand still.
In crash bandicoot team racing for Xbox one you can drive as a female character in the track named "Tiny course" or something like that, it has a lot of mud puddles, you can just stop your cart in the mud or pas really slow and see how the mud covers your character, I do it with coco bandicoot, her jeans and white shirt would look so beautiful covered with mud
In the first dead rising one of the weapons you could find was pies and you could hit any npc in the game with a pie in the face and they'd react, wipe their eyes, and would have coverage and everything. It was awesome. I tried to report it once or twice here but never got any replies.
I did some checking and found that Disney Infinity has some messy content. Particularly, in the Inside Out playset. There are pie launchers in that set that will get your character coated for a moment should they hit. I have no idea if there are other instances, though.
6/14/22, 5:38pm: This post won't affect thread last post date.
Me and some fellow sploshers have been working out a mtg based Wam card game.
We'd basically re-skin mtg cards with new wam based cards. For example we'd rebrand all mana as a different messy substance. Like white mana would be pie mana and green mana would be gunge mana.