Probably missing some but I think you get the point I'm making here.
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The list is huge, because a lot of champion can initiate fights. It basically comes down to your team composition and strategy.
Some initiate by locking down one guy (e.g. Maokai, Leona); some disrupt single champion/entire team positioning (e.g. Gragas, Orianna); some shut down the whole enemy team outright (e.g. Amumu, Galio).
You have hard initiators, who basically go in and will probably survive the initial onslaught of enemy attacks (e.g. Malphite, Alistar, Singed). You have soft initiators, who will need immediate protection and/or escape option after they make the first move (e.g. Morgana, Cassiopeia). Then there is a mix too. For example, Sona can play hard suicidal initiator.
Finally, you get indirect initiators like Shen (who can initiate on his own anyway) and Zilean, who can allow practically anyone to initiate by offering some form of death prevention.