I was thinking about this question and thought of maybe printing a pattern-drawing roller painter.
The question is: is it possible to print with an ink absorbing material that could make a paint roller possible?
P.S: I don't own a 3D printer, nor have I any deep knowledge in this matter. I simply want to know if this is feasible, so I can start looking for someone to 3D print this for me. If it's not, knowing beforehand could spare me a lot of time.
While printing a roller stamp or rubber-roll from a flexible material such as [hard]TPU, [softer] TPE, or even a [super soft] foaming flexible filament is certainly possible. In any case, this would create soft, somewhat squishy prints. These prints will work quite easily as a stamp or woodblock printing stock, transferring ink from a pad to paper. The print pattern will depend a lot on how soft the stamp is: the harder, the sharper it will print. A massive roll of this material can behave akin to a rubber roll as one uses it in linoleum printing.
A foaming filament might be able to take a little paint in its airgaps, but it will never be as soft and contain as much paint as a foam lacquer roll - making it at best an improvised tool, or one that is chosen for a specific artistic purpose. It behaves more akin to closed-cell foam, while foam brushes and rolls from artist supply are typically open-cell ones. So if you go for a roller-stamp, you'll need to have an ink reservoir in the shape of a soaky-roller that isn't printed.
To top it off, it is nearly impossible to print a hairy wall painting roller: the hairs used in them can't be achieved with common print materials and slicers at the time. Even if stringing creates hair of similar dimensions, they are not affixed well enough to not get lost in the paint and can't be reliably created on demand.
Notes on pricing: One of the few options for foaming print materials is colorFabb, who was the first to offer such. Most of their foaming filaments cost around 50 €/kg, their flexible NinjaFlex costs about 80 €/kg.
It's possible to 3D print in materials like TPE that are rubbery. In theory, one could print a sheet of the pattern and then wrap that sheet around a roller. That would be expensive, though, and I doubt TPE would absorb enough paint/ink to lay down an good and even coat.
As pointed out in the comments,there is one of foaming TPU filament (Varioshore TPU that might be able to achieve the kind of soft, spongy feel that a paint roller would need but it's expensive and, I suspect, soft but not particularly absorbent in the way you would need.
I have no direct experience with it, though, so I can't say for sure. For the price and amount of time that would be required to get a print roller produced the way you need, I think you'd be better off buying a custom pattern roller/brayer or making one yourself.
I imagine you could achieve this by getting a sheet of stamp pad foam and plotting it with a CNC cutter (or cutting it by hand) and then taping/gluing it around a tube.
It might be possible to directly print something like this, however it might not be the easiest method. Let me unpack that a bit.
The issue for 3D printed foams is that, even the low density foams, such as the ones from ColorFabb, still are fairly high density on the scale of foams. (they are about 50% as dense as solid filament when expanded. They are also great materials, side note, just not for this application) They are also what is called "closed cell" which means, while they have hollow spaces inside them, those spaces are closed off from each other. This makes them not very useful for things like holding paint, because the effective surface area is not substantially different from a solid part.
There are alternatives, materials out there which print solid but contain a fraction of soluble material. So you print, and then post process to dissolve away those elements, resulting in a high-porosity filament. Something like Poro-lay might work. (you can find it for purchase HERE)
There might be an easier path here though, and that would be to use 3D printing as an intermediate step. 3D print the design of the roller you want, and use it as a mouldmaster. Pour a silicone or urethane mould material over it, making yourself a mould into which you can then cast any of a wide variety of different rubber or foam materials to make your roller. 3D printing as an intermediate step, rather than end-use-part, is very very powerful in that way.
With all of this, chemical compatibility between your paint and end material should be kept in mind. Some paints contain solvents which will attack various foams, plastics, etc.
Finally, while this is digressing from the realm of 3D printing, there is one more solution I might consider exploring. A C02 laser will cut, or engrave, the felt on most paint rollers. If you can find someone with a rotary axis who is willing, they can simply cut/engrave the felt/foam of a factory paint roller into whatever pattern you like. (this is how the CUSTOM RUBBER ROLLERS linked in @Rykara's post were made) It wouldn't actually be terribly difficult, however you'd probably want to experiment on a couple different rollers to ensure the effect is what you want.
I hope that helps. :)
I'm relatively new to 3D printing, but I know some stuff. If you really wanted to print a paint roller with a pattern, I would actually go with nylon or TPR. TPE is okay, but I find that it is less cushy. I would highly suggest buying your own printer for this project. It would be much more cost effective in not only the long run, but for testing different materials. 3D printing services get pricey, so choose your printer wisely. If you do decide to buy your own printer, I would go with an Ender 3 pro. Mine is very smooth, and has an easy-to-use interface.
