How much will I be missing out on if I don't have
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Online gave you access to the city which meant you could play with other hunters and eventually get to high rank and complete the upgrades for your weapons and armor also for 3 online only monsters Devil Jho, Jhen Mohran and Alatreon. Also event quests with special weapons and armor.
Online Multiplayer, Loc Lac City, and Event Quests are the only things you won't have access to anymore. Everything else that has to do with online only features have to be accessed through WiiRD (Essentially a game enhancer like gameshark, action replay, and codebreaker) and mods which both requires you to softmod your Wii.