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Is it possible to increase the pixel density of an image?
Is there a general formula for image size vs. print size?
Can one of the many photo editors max a photo automatically while preserving the same quality?
The answer is the max size a photo can be enlarged is the size of the original photo.
No. Photo editing software cannot recover image detail that is not there in the image originally captured at the scene by your camera. So, enlarging the photo to a very large size will eventually reveal a lack of detail.
Apparent print quality depends a lot on the combination of print size and viewing distance. If you take your photo, print it A2-size and view it from 10m, it will look good. If you walk up to it and view it from 0.2m, it will look worse. No matter what the resolution of the photo, pretty much.
To state this again differently, printing the photo larger will always generate a worse looking photo, unless you also increase the viewing distance.
For standard viewing distances, around an arm's length perhaps, many people find around 300dpi sufficient. Meaning, if the image is 1" x 1", it needs to be 300 pixels per side. If you double the distance, you can halve the number of dpi you need. If you double the image size, you will need double the number of pixels per side for similar quality.
The camera I use most produces images at 4288*2844. At 300dpi, I can produce a print as large as 14" * 9.5" before I start to notice falloff in quality at normal viewing distances (that is, 4288 / 300 = 14.293333... which I rounded to 14 inches).
There are image editing plugins (for Photoshop for example) that help you resize a photo with good subjective quality by inventing image detail that really wasn't in the original photograph, but looks believable, and doing things like Anti-Aliasing to reduce the incongruous appearance of the magnified pixels. Alien Skin Blow Up 3 is an example, but I have never used it.
As James Youngman points out, it depends on the viewing distance. It also depends on the printer resolution - this is somewhat subjective, but I would imagine you'd want to print between 200-300dpi in most cases.
If you assume a 300dpi print resolution, then simply divide the pixel resolution of the height and width of your image by 300 to get the number of inches you can print.
Resolution @300dpi @200dpi
6MP 3000 x 2000 10x6" 15x10"
8MP 3500 x 2300 12x8" 17x12"
10MP 3900 x 2600 13x9" 19x13"
12MP 4200 x 2800 14x9" 21x14"
14MP 4600 x 3100 15x10" 23x15"
16MP 4900 x 3300 16x11" 24x16"
18MP 5200 x 3500 17x12" 26x17"
Here is how to calculate this yourself:
To estimate the pixel dimensions for a number of megapixels, assuming a 3:2 aspect ratio, divide the megapixels by 6, take the square root, then multiply that result by 3 for your width, and by 2 for your height. For example, a 14MP camera. Divide 14 million by 6, which equals 2.3 million. Take square root of that, which equals about 1500. Multiply 1500 by 2 and by 3 to get your dimensions (3000 x 4500).
Now divide 3000 and 4500 by the dpi (for example 300dpi to arrive at a print size of 10" by 15")
