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You can only have three quests available at any given time.
A few things to note regarding these:
When "quest 4" would have come around, it does not erase an old quest to make room, you continue to have the three prior and none "queue" up.
Everyone is given a random quest each day, meaning that your friend and you could have got two completely different quests.
You cannot have two completely identical quests at the same time. i.e. You'll never have two "Play Diablo characters" quests, but you could have a "Play StarCraft characters" and a "Play Diablo characters."
You can progress through multiple quests at a time. So if you had a 'Support' and 'Warcraft', you could play Lili and make progress towards both.