For example, a hand of bridge contains 13 cards. A hand of machiavelli contains 15 cards. Do games with even larger hands exist?
Many games do not have a hand size limit but are functionally limited by the number of cards available. It would be possible, for example, for someone to draw all but two cards in a game of Uno (the other player must hold one, and one must be in the face-up play pile). A standard Uno deck has 108 cards, so it would be possible for one player to hold a hand of 106 cards.
There isn't really any reason why you couldn't play Uno with multiple decks shuffled together, however, in which case the upper limit of the hand size is just dictated by how many cards you're playing with.
Sergeant Major card game deals 16 cards to each of the three players.
First deal and exchange
Draw cards to determine the first dealer and deal 16 cards to each player, the 4 remaining cards are placed on the table to form a kitty.
Any three-player game using a traditional 52-card pack may be dealt 16 cards each.
Another example is rummy where a player can hoard as many cards as they please during the play.
In Magic the Gathering, there is no upper limit. (You are practically limited by the number of cards in your collection.)
Hand limit is only enforced at the end of your turn, and even this can be circumvented with cards like Spellbook. While your deck is limited to the number of cards you can shuffle unassisted, in a non-tournament game you can gain access to an unlimited number of cards through effects like Research. Research has the card specific ruling:
A card “from outside the game” may be a card from your collection or a card from your sideboard. In tournament play, you can’t choose a card from your collection. The cards you choose don’t all have to come from the same place.
So, outside of a tournament, you can use Elixir of Immortality, Staff of Domination, Research, and any infinite mana combo to get nearly all of the cards in your collection into your hand (you will be missing the cards on the battlefield powering this combo). That can easily be thousands of cards.