A normal distribution.

Levy stable [0].

If I had made the extra condition that the random variables had finite variance, you'd be correct. Without the finite variance condition, the distribution is Levy stable.

Levy stable distributions can have finite mean but infinite variance. They can also have infinite mean and infinite variance. Only in the finite mean and finite variance case does it imply a Gaussian.

Levy stable distributions are also called "fat-tailed", "heavy-tailed" or "power law" distributions. In some sense, Levy stable distributions are more normal than the normal distribution. It might be tempting to dismiss the infinite variance condition but, practically, this just means you get larger and larger numbers as you draw from the distribution.

This was one of Mandelbrot's main positions, that power laws were much more common than previously thought and should be adopted much more readily.

As an aside, if you do ever get asked this in an interview, don't expect to get the job if you answer correctly.

[0] https://en.wikipedia.org/wiki/L%C3%A9vy_distribution