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I like the analogy of a snowball, nothing ever enters it yet it grows.

This accurately describes how I gain weight as well.

This joke is in direct violation of the First Law of Thermodynamics.

Yes, but the mass does not have to cross the event horizon for that, does it?

At that point we probably can't tell more without diving into the math, and it's beyond my abilities. Yet the gross idea does not seem absurd to me.

Yes, the radius of its event horizon is directly proportional to its mass. The relationship is described by the Swartzchild equation.