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The 'Big burger' is the 7th enemy you can fight on the Monsters tab in the Fight section. It's roughly 6 orders of magnitude stronger than the first enemy (Slimy) so you'll have to build up your Battle stat by a fair bit.

Things that increase your Battle stat:
- defeating Gods
- defeating (lesser) monsters
- Physical training
- building monuments (though you probably haven't unlocked those at this point)

Rebirthing will speed up your progress. I was able to defeat my first Big burger when my multipliers were in the 10,000 range. Just start killing stronger and stronger monsters as you're able, when your battle stat increases. Don't try to jump ahead and dump a thousand clones into battling a Big burger if you're not strong enough yet.