<p>I have a 3D printed object that I'd like to print on. Adhesive stickers are an option, but as the surface is rounded it's difficult to get a good film to stick well. </p>

<p>Is there any system to print on a 3D object (e.g. ink jet). I need at least 300dpi. Black in the first instance, but color would be nice for future projects. </p>

There are printer types that can print images into the material as it's being printed. SLA and SLS type printers are capable of this but I believe Polyjet printers are the best suited.
At 300 dpi though you might be pushing the edge of what is capable. to get that fine of detail your best option may be to look into getting a vinyl wrap for your part that is applied after it's printed. If you've seen cars with really high detail images on them, a vinyl wrap is typically how they're done.
Note
I should state that the SLA and SLS embedded images are still in the R&D phases and being experimented with. Possible now, but not in production phase. The SLS version essentially ran another print head over that worked similar to an ink jet before sintering the layers. The SLA version I saw worked somewhat similarly, but had some 'unique' clearing processes.