There's also protect the carry style comps where you go with a largely supportive team shielding/healing/reviving/peeling for one hypercarry. I think you already listed most of the popular ones though.
Lane/Control: Heavily focussed on gaining advantage in the laning phase and snowballing as a result.
Global: Using multiple sources of global presence to gain control of the map - a subtype of the Lane/Control comp.
Rush: A comp built around rushing Towers and objectives; getting fed on global gold to snowball.
Late-Game: A comp that wins by default in late-game. Involves multiple hypercarries such as
Ryze and
Vladimir or
Vayne and
Karthus.
Global: Using multiple sources of global presence to gain control of the map - a subtype of the Lane/Control comp.
Rush: A comp built around rushing Towers and objectives; getting fed on global gold to snowball.
Late-Game: A comp that wins by default in late-game. Involves multiple hypercarries such as





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I know about Poke, Split-Push, Dive, and AoE-Burst comps but that can't be all there is.