Final answer:
Hydrogen and helium are the two atoms that have their outer energy levels filled with 2 electrons; hydrogen has one electron and helium has two, filling its first shell completely.
Step-by-step explanation:
The two atoms that will have their outer energy level filled with 2 electrons are hydrogen and helium. This is because they are the only two elements in the first row of the periodic table, which corresponds to their having electrons only in their first shell. While hydrogen has one electron in its outer shell helium has two therefore completely filling that shell according to the content loaded within our resources.
This full shell is essentially following the Pauli exclusion principle, which dictates that no two electrons in the same atom can have the same set of four quantum numbers which is why the two electrons in the helium atom have opposite spin quantum numbers.