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Human genetic code has maybe 0.2% variation in base pairs, the average individual gene has maybe 13 thousand base pairs, and the whole code has about 3.1 billion base pairs.
This is crappy, quick and dirty math, but this might be one of those high-complexity situations (like weather modeling) where minimal detail can be closer to the truth than an intermediate amount anyway. So, we'll model it as a random selection of 0.2% differentiated genes, meaning ~240,000 take ~500, which comes out to a number with 1556 digits. The amount needed to get a 50/50 chance of a repeat will be lower, with 777, almost 778 digits.
If you allow weird, implausible and mostly stillborn combinations it's closer to 2^~240,000^ . If you include mutations, it gets much much higher to the point of basically being undefined. It's more about how many changes is still human, and after that how big a genome a cell can handle.
My go-to insult from now on is going to be:
"6.213165126500310316378124236349955462389737376162847388361.. x 10^1555^ possibile human specimens, and somehow I'm unfortunate enough to have to come across you."
Awesome answer! Thanks.