I was wondering how would you print an image that's scaled three times its original size without making it look like crap? If you change the dpi to 300 and print it'll look like crap. Is there a way to convert it gracefully?
You should vectorize your image, scale it, and if you wish you may convert it back to the original format (jpg, gif, png...). However this works best for simple images. Do you know how to vectorize? There are some sites that do it online, just do some Google research and you'll find some.
Changing the DPI won't matter if you don't have enough pixels in your image for the size you are printing. In the biz it's called GIGO (Garbage In, Garbage Out).
If your image is in HTML then create a media="print" stylesheet and feed a high-res image that way.
You may have the problem of trying to add detail that isn't there. Hopefully you're aware of this.
The best way to enlarge an image that I know of is to use bicubic interpolation. If it's any help, Photoshop recommends using 'bicubic smoother' for enlargement.
Also, be careful with DPI vs PPI.
This is called supersampling or interpolation. There's no 'perfect' algorithm, since that would imply generating new information where there was none ('between' the pixels); but some methods are better than others in fooling the eye/brain to fill the voids, or at least not making big square boxes.
Start with the wikipedia articles on Nearest-Neighbor, Bilinear and Bicubic interpolations (the three offered by PhotoShop). A few more Tricubic interpolation, Lanczos resampling could be of interest, also check the theory, and comparison links.
In short, this isn't a cut-and-clear issue; but an active investigation field, full of subjectivity and practical trade-offs.