Sokoban

The game you played a lot, without even knowing about.
Sokoban 倉庫番 means storekeeper or "warehouse man", the object of the game is to push crates to predefined destinations.
Sokoban is a puzzle game that has been used, in some form or another, as a minigame in a number of different games, predominantly jrpgs.
Sokoban was created in 1982 by Thinking Rabbit for the NEC PC-8801, a popular Japanese home computer, and got a sequel in 1984.

Original Sokoban for the NEC PC-8801

The game was such a hit that the term Sokoban became synonymous with this type of puzzle; not only the original games made by Thinking Rabbit.
It has been ported to various platforms, and is common as web games and apps and as minigames in a number of other types of games, but often altered and reinvented.
One example is the "Millenium Puzzles" in the 2002 PS2 game Wild Arms 3, where it was turned into three dimensions.

Millenium Puzzle in Wild Arms 3

Love it or hate it, Sokoban has certainly left its mark in gaming history.