If you have a HP printer the Photo RETII or RETIII mixes the colors in single dots and allows a lower dot count with very little difference in output quality. I can print 1200 X 2400 or RETIII on premium photo paper. HP doesnt tell you but I think RETIII is about 600 DPI. RETIII spools faster and prints a little faster with little discernable difference in quality.
You have to use premium photo paper to print at higher DPI. Plain paper will not take the ink and quality goes down rather than up by trying to print to plain paper at too high a resolution. With most printers the only way to get it to print at too high a resolution for the paper you are using is to tell it you have better paper not a good idea.
If you are talking about input resolution it is a different ball of wax. The DPI you get from an image editor or scanner is pixels per inch. A modern inkjet makes over a dozen dots to represent a pixel at best photo quality. So 300 DPI from your image editor or scanner is over 3000 inkjet DPI, which is higher than any current model will produce. You are just spinning your wheels feeding more than 300 DPI (in pixels per inch) to any printer. The printer spooler just has to downsize the file so it can use it.
Photoshop is the high end photo editor of choice for most professionals. It wont send a print job out at over 300 DPI no matter how high the resolution of the image being printed. At least that is true through version 5.5. It is sensible as no printer on the market can use more than that.