Just how far can I progress?

Okay, so, I finally found the answer.

The answer:

While you can't strictly stop progressing in the game, there does come a point along the line where the game breaks; after each click you will gain a level, and the progress bar overflows in appearance past its limit on some clicks. This is because when you reach this level, the experience required for your next level goes from 999999999956753000000 to 1.0000000000199987e%2B21; the number stored in the experience_for_next_level cookie gets expressed in terms of an exponent in order to prevent the number from growing too large numerically. The game, however, is unable to parse this value, and so it gets cut off at the 1. Consequently, you gain a level each time you click on the progress button.

The lowest level at which this happens is 31622776601. The last level of the game before the game becomes broken is 31622776600.

How I learned to get this value and love the bomb:

I installed a cookie editor add-on for Firefox and messed around with the level cookie for about 30 minutes to figure out the highest possible level. I would start at level 1, change the level cookie value to some arbitrary amount, refresh, click to level up(this increments my new level by 1 and also causes the game to go through and edit all the other cookie values, such as experience_to_next_level, based on my new level. After each edit, I would delete all my cookies and refresh the browser to restart at level one. By this procedure I was able to experimentally determine the lowest level at which the game breaks.