The amount of experience to hit the maximum level in-game (30) is 241,801 exp. However, the experience counter doesn't stop there. I have gone as far as about 400,000 in-game experience, and I'm wondering how high can the number go. Is there any cap?
No, there is not. There is even a trick to get over one million experience with the help of teammates (in AI game). On the map "Towers of Doom" - if you defend levers and destroy enemy forts, leaving them always with at least one fort (not to make your nexus start shooting the enemy one), you can grab quite a lot of experience.
As of early 2018, the cap has been discovered. The maximum base experience one can gain in a single game is 500,000, which when multiplied by Friend Bonus + Party bonus + Stimpack Boost gives the maximum of 1,375,000 experience. With the First Win of the Day Bonus, it gets multiplied to a further 1,625,000.
However, if the in-game experience goes over 520,000, then an overflow happens and the experience awarded will be set to 0. This is true for all values above 520,000 as well.
Thus, for people trying to maximize experience, try to start ending the game around 475,000 experience as it racks up relatively fast in the late-game. Remember that the Core itself also grants experience!