If you have only one exposure, your options are limited to upsampling or using software to create more pixels. While the information rendered by software is never as rich and sharp as information that’s optically captured, it nonetheless can be both pleasing and convincing. Upsampling is the best method for images containing moving objects, as other methods require multiple exposures and may produce motion artifacts.
Upsampling with Lightroom or Photoshop works well for most situations. Use the upsampling method of Bicubic Smoother (best for enlargement). It’s as simple as that. Upsampling in Lightroom can be performed manually during exporting or automatically during printing. If upsampling is substantial, I prefer doing it in Photoshop where follow-up sharpening can be better fine-tuned. In Photoshop, go to File > Image Size, and check Resample Image. Set your desired Pixel Dimensions or Document Size; they’re two different ways of asking Photoshop to do the same thing: resample an image. Pull down from Bicubic Automatic to Bicubic Smoother, which is best for enlargements. (You don’t actually have to do this step as Photoshop automatically chooses the best resampling method.) Click OK.
Much has been made of "stair-stepping" methods, where files are increased in size in small increments (sometimes 25%, sometimes 10%) multiple times. However, the increase in quality gained is minor, making this rarely worth the effort. The thing that makes the biggest difference in image quality is the sharpening applied to images after they have been upsampled. If you want to maximize quality, after upsampling, carefully sharpen images individually. Furthermore, if you’re upsampling an image dramatically, before upsampling turn off all prior sharpening and noise reduction, and after upsampling apply new sharpening and noise reduction. Upsampling routines will amplify noise reduction and sharpening artifacts, which may produce inferior or even objectionable final results.
If you’re performing extreme upsampling, consider onOne Software’s Perfect Resize. It can make a poster out of a postage stamp. Perfect Resize produces superior edge contrast that’s very useful when you’re aggressively upsampling. If you’re performing moderate upsampling, you won’t find the differences between Photoshop/Lightroom and Perfect Resize to be dramatic.
How far can you go? I could give you an overly simple answer: up to 300%. But the true answer is, it depends. Knowing what it depends on will help you choose a method, modify a routine and evaluate results to get optimal results for individual images. How far you can upsample an original depends on many factors found in the source, the destination and the statement you’re making.