Rank-up test

From FFXI Wiki

Once you reached the maximum skill level for your current crafting rank, you can no longer receive skillups. You need to pass a rank-up test in order to reach the next rank. It is possible to trigger the test two levels below your current rank skill level cap.

To trigger the rank-up test, talk to the master of your crafting guild. He will ask you to bring him a specific item. Trade him the item he asked you for, and he'll congratulate you and will raise your rank. With the exception of Veteran rank, the item you trade him can be crafted by you or anyone else, or bought from the Auction House or a Bazaar. To raise from Veteran rank for all crafts (except for fishing), you must trade an item that is inscribed with your name. All versions of the required item (NQ, HQ1, HQ2, HQ3) are accepted by the guild NPC.

You can see the list of the rank-up test items on every crafts page Alchemy, Bonecraft, Clothcraft, Cooking, Fishing, Goldsmithing, Leathercraft, Smithing and Woodworking.
Alternately, you may find the list of items on every rank page:

Ranks
Amateur 1-10
Recruit 8-20
Initiate 18-30
Novice 28-40
Apprentice 38-50
Journeyman 48-60
Craftsman 58-70
Artisan 68-80
Adept 78-90
Veteran 88-100
Expert 98-110

There exist several ranks in the game's data files that are not implemented in game. These are Authority, Luminary, Master, Grandmaster, and Legend.