In the mining battles there's a hint stating that there is a 'highest' composite. I imagine that this would be achieved by fighting an opponent with lots of energy, like maybe 97, having it steal a 13, you play 13 at it for the full 300 and then open up a single tile for it using the tongs, it plays 13 back (and ideally ONLY plays that 13), then you 13 again and also 43, and play your element x which would be worth possibly as high as 918, which would be 2*17*3^3. 

Also, it may be worth checking for squares of some of the higher primes that are normally impossible to square. Like how 7 has a unique effect when squared, maybe other high ones would too? Or 7^3? Input 2.0 combined with the previous 13 trading would probably have good odds of creating some of these.

Okay, I've been labbing it out in the fight with 13 which is pretty consistent at generating tons of energy. 169 (13^2), 289 (17^2) and 361 (19^2) don't seem to do anything special. I at least thought that 19^2 would share 11^2's property of increasing the area of effect, but no, the range is still just 1 tile orthagonally. 343 (7^3) also seems to just do the standard 49 effect. It should be stated that it's possible these do have unique effects, and I may just not be meeting the conditions to even notice them. Like maybe 49*7 would have some further expanded cleansing effect that I am just not noticing because I am not being hit with the right effects to be cleansed.

Wow, even 527 (31*17) doesn't allow diagonal tile removal. That one especially is surprising. 

323 (17*19) is also lacking the diagonal removal effect. The only other one I can think to try is 23*17 to see if you can place two diagonal tiles away.