How to automatically pull text out of a video I am currently experimenting with after effects and premiere to see what sort of strange things they might be capable of and was hoping someone had some insight on an Idea I had.

In this situation I have some text, such as some subtitles or even a time code, that is burned into the footage. I do not have any of the project files for this footage, the only thing I have to work with is the final video.

Is there any way to have either Adobe Premiere or After Effects recognize what text is on screen automatically? Bonus points if it can then take this information and feed it into a new title element. 

P.S. This is all assuming the source video's text is of decent quality (i.e. the text is static, decently distinguished from the video behind it, and not too tiny.)
I do not know of any existing solutions for this in Premiere or AE.

However, if you know a little about programming, you might be able to write a script using OpenCV that runs Tesseract OCR on video frames, and comes out with the text you want. Once you have the text and time codes, you definitely could write a script to build the titles at the appropriate time codes inside of AE or Premiere.

(I became aware of these open-source tools since I am currently building a mobile app that runs OCR on documents.)

Check out the Scene Text Recognition docs for OpenCV, if you want to try that route.