When Lichess analyzes a game, how deep does it analysis? In reference to the request computer analysis feature, how how deep is Stockfish analyzing, such as looking 12 moves ahead?
It tells you at the top right. 



The first number (19) tells you the depth it has analyzed to (19 ply = 9.5 moves). Note that because of engine heuristics, it doesn't exhaustively search for every single move 9.5 deep. The second number (23) tells you the max depth to which the engine is searching for. After the bar fills up, there is an optional feature that lets you increase the depth beyond 23 if you want to explore a specific position further.