In Magic: The Gathering, there are Draft and Sealed tournaments that are very popular. Heartstone also has this type of game mode. It helps to spread interest in the game, and players sometimes get to keep the cards that they opened during the tournaments (for MTG). Sometimes the entry fee is basically the cost of the packs. So essentially these tournaments are a fun way to buy more packs. The company that creates MTG (Wizards Of The Coast) often supports prizes for the winners.
Illivium could have similar modes. They would have an entry fee, and participants would all keep their Illuvials, with winners getting extra prizes.
EXAMPLES:
DRAFT
MTG's Drafts: Players sit around a table (tournaments can have multiple tables) and each player opens a pack of cards. Each player takes a card from their pack, and passes the pack to the next player. Repeat until the packs are empty. Then repeat for 2 more packs. Then players make the best decks they can with the cards they drafted, and play a tournament.
Illuvium-Style Drafts: Players pay an entry fee and enter a lobby. When full, everyone is given a "pack" of 15 random Illuvials. Players pick one and pass, until the pack is done. Repeat with 2 more packs (or maybe significantly more, to allow players to accumulate duplicates and fuse them to higher-stage Illuvials). Then players play an 8-player tournament with their decks. The winners get additional prizes in the form of Illuvials or Fuel.
SEALED:
MTG Sealed: Players each get 6 packs. They build the best decks they can with those cards, and have a tournament. All players keep the cards they opened. Winners get prizes of additional packs.
Illuvium-Style Sealed: Players sign up, pay an entry fee, and get a week to do a set number of Overworld runs (maybe 10 or 20 - depends on how long a run is and how many Illuvials can be caught). At the end of this time period, players build the best deck they can with Illuvials caught, and then have a tournament. Players keep Illuvials they caught, and winners get prizes.
Why These Are Great
These are isolated tournaments where everyone starts on the same playing field. Anyone can win, if they build a good deck and play it well. Skill matters, but a smart new player can beat a whale. So people love to "buy" packs by participating in these tournaments. The result is that way more packs are purchased, and the fun of new sets last months without players getting bored.