What is the neutral atom that has its first two energy levels filled, has 2 electrons in its third energy level, and has no other electrons?

The element would be Magnesium.

Explanation:

Magnesium has 12 electrons in total.

The first shell (energy level) can hold 2 electrons.

Then, every shell after this can hold a maximum of 8 electrons. So, a full second shell would take us up to 10 electrons. The spare 2 are in the third energy level as stated in the question.

(2+8+2 = 12)

The electronic configuration for this can be written as 2.8.2 or 1s^2 2s^2 2p^6 3s^2