The single, greatest online experience I ever had and will only, probably, be toped by it's successor, is without a shadow of a doubt, Diablo II LOD.
Nothing in my 15 years of online gaming has come close to the experience that the countless hundred, if not thousands, of hours in that great game have provided to me.
Since the OP also asked a console counterpart in the case of us posting a PC game, currently, the best online experience I've had is Demon's Souls. The entire game is an online experience unto itself, with it's functionalities very well executed within the game.
Current PC Build
CPU - i7 8700K 3.7 GHz (4.7 GHz turbo) 6 cores OC'd to 5.2 GHz with Watercooling (Hydro Series H110i) | MB - Gigabyte Z370 HD3P ATX | Gigabyte GTX 1080ti Gaming OC BLACK 11G (1657 MHz Boost Core / 11010 MHz Memory) | RAM - Corsair DIMM 32GB DDR4, 2400 MHz | PSU - Corsair CX650M (80+ Bronze) 650W | Audio - Asus Essence STX II 7.1 | Monitor - Samsung U28E590D 4K UHD, Freesync, 1 ms, 60 Hz, 28"
