1. e4 e5 2. Nf3 Nc6
This is called the King's Knight's Game.
Most popular at the top level since the top level existed has been the Ruy Lopez. 1. e4 e5 2. Nf3 Nc6 3. Bb5 and black will often respond 3. ... a6 where white has two very viable options in Ba4 (most popular) and Bxc6 which is also interesting.
Which is why I said Bb5 instead of Bc4 :) ... if you don't want 2. Nf3 you're gonna need to 1. Reword the question, and 2. play f4 :).
The chess.com explorer shows this for the five most popular responses to 1.e4 e5.
move | games | white wins / draw / black wins | |||
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2.Nf3 | 297,572 |
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2.f4 | 16,675 |
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2.Nc3 | 16,487 |
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2.Bc4 | 13,658 |
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2.d4 | 6,419 |
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There is more here.
King's Gambit 1.e4 ef 2.f4
Vienna game 1.e4 e5 2.Nc3
No name 1.e4 e5 2.d4
That's a lot of information you can use to determine your next move.
Anybody got an opinion what to play against e4 e5 apart from Nf3 Bc4
Or any white ideas