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The video world is always 72 dpi. Rick will probably chime in indicating that there is no "real" dpi in video at all. 3D programs have the ability to render out imagery at higher resolutions, but this is not an option here. If you need more pixels (which in turn would give you more detail when cramed into a smaller space), then your assumption about having to enlarge your comp is correct. There is a script that ships with AE called "scale comp" (you can find it under File > Scripts > Demo Palette) that will allow you to alter your comp size by either percentage, or speicific value. I would try that route. If all of your content was created from within AE, your quality will probably be fine at the end of the enlarging procedure. If you imported raster artwork, or footage, then that quality will suffer during the enlargement procedure.
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Actually the video and web world doesn't have a dpi or more accurately ppi (pixels per inch because dots refer to line screens in the offset printing industry). There is no size tag in a gif file or in a video file that defines PPI so image editing apps assume 72, which used to be the resolution of the average computer monitor in pixels per inch. After effects does not add a PPI tag to the image headers so image editing apps (Photoshop for example) assume 72 PPI.
Modern displays have a much higher pixel density so the number 72 is basically meaningless. IOW if an image is 720 pxels wide and you're viewing it at 100% it is no longer 10 inches wide on the computer screen, but it will still be 10 inches wide when you send it to a printer. The only time PPI means anything is when you send the image to a device that will print it.
Now that your thoroughly confused let me throw another wrench into the gearbox. Resolution has nothing to do with PPI or DPI. High resolution images have a lot of information (pixels), low resolution images have a low number of pixels. You can set the PPI of a 100 pixel X 100 pixel jpg to 9999PPI, but it will still only contain 1000 pixels. You can set the "resolution" of a 30,000 X 30,000 pixel image to 1PPI and it will still be high resolution because it contains a bunch of pixels. Nine hundred million to be exact.
That said, if you originally have a composition that is small and you need to output something that is bigger then the first thing to do is figure out what is the minimum resolution required to print the image at the quality you desire. If you need a final image that is 11 inches wide to be printed by an offset press then you probably need 11 X 240 or a composition that is 2640 pixels wide. Providing the composition only contains vector objects (shapes, masks, type, or illustrator files) you can resize the composition using the Resize Comp script that comes with AE. You'll find it under File>Scripts>Sample Scripts.
You may also be able to nest your original composition in a larger composition, collapse transformations, then scale it up. If the original composition contains low resolution images or video you will not have as much success scaling things up.
One last thing, lots of folks think that 300 PPI is the minimum for printing. That's not so. The optimum resolution for printing is determined by the type of printer, the viewing distance, the material. Those great big full color banners you see at the mall are probably printed at 50 or 72 or maybe even 90 PPI,
40" posters typically look great at 150 PPI, while glossy waterless offset business cards may require 600ppi for the type to be clean.
I hope this helps.
I know there are photoshop plug-ins such as Genuine Fractals.
500% is way beyond what even the fanciest and most sophisticated "pixel invention algorithm" can make look good, especially if it needs to be printed. It will only look noisy and the reduced gamut of CMYK space will result in wrong colors all the way, ink density problems and what have you. Your only option is to rebuild everything cleanly in Illustrator or Photoshop, starting out natively in CMYK.
Thank you very much for the responses. I was not aware of the File>Scripts elements and since I did not use any raster elements (thank goodness) I was able to use those to adjust my composition and element scales. Everything is looking good so far. Also thanks for the tip so I didn't have to spend money on the photoshop plug-ins. I was hoping to avoid them at all cost.
I found these suggestions helpful too for a project I have going on.
On a separate tangent, I wish AE could export a vector AI file.