An amount much larger than N but smaller than infinity. "I have 2-to-the-N things to do before I can go out for lunch" means you probably won't show up.
Numbers of the form two-to-the-N are very important in computing because they represent the value of bit N of a binary number (counting from 0) and the number of things you can count with an N bit number.
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Nearby terms: twonkie « two-phase commit « twos complement « two-to-the-N » two-valued logic » TX-0 » TXL