Here are a few that have come up in my saga:
I have cut out the ones listed in the Rhine Tribunal book, to avoid copyright issues, but I have included challenges for some magi in both the Rhine and Normany books who did not have challenges listed.
The list of number of times attempted and passed does not reflect PC attempts.
Provencal Tribunal
Archmagus Karl Vertoff of Bonisagus (Doissetep)
Penetrate his full magical defenses with a fifth magnitude (or higher) spell that will not do him real harm. (He knows a point of warping may result, but is willing to accept this).
This challenge has been attempted a reported six times. One applicant was successful.
Archmagus Oxioun of Tytalus (Doissetep)
After formally making the challenge the applicant has one moon to steal his talisman and keep it from him for one full moon. For the duration of the challenge Oxioun will not slay the applicant for violating his sanctum, but he may use non-lethal means of injuring them while they forfeit immunity.
This challenge has been attempted three times and never successfully completed.
Oxioun is not known to HAVE a Talisman (though he must if this is the challenge). In one reported case the applicant managed to get into Oxioun's sanctum and was placed in a magical, season-long sleep.
Archmaga Kerka of Merinita (Doissetep)
Kerka's challenge is to best her and her storm elemental familiar in a race, without the use of teleportation. The race is from a temple of Zeus outside Athens to the pillars of Hercules. During that time your feet may not touch land nor sea.
This has been attempted twice, neither successfully. The race takes her about four days.
Normandy
Archmaga Geirlaug of Flambeau (Montverte)
She does not seem to have been challenged in some time. Access is reported to be difficult.
Her challenge is a battle in the style of Dimicatio (L&L p. 32)
This has been attempted ten times, and three have succeed.
Archmagus Mathieus of Constantinople of Jerbiton (Eboris)
Demonstrate a mastery of urban spellcasting. The precise challenge is different for each magi, but it often involves casting multiple subtle spells (which do not breach the code) within the great cathedral of Paris. These spells are often harmless illusions but they must overcome the aura, and be cast silently and without gestures.
This has been attempted thirty-one times (at least) and has been successful once.
Rhine
Archmagus Igor Rastvan ex Miscellanea (Radhost)
Using a ritual spell, and without help from any other gifted person, slay a condemned prisoner (slave) located in the courtyard of the temple of Razhost. The date and time of the challenge must be agreed on. This must penetrate the Aegis of Rzhost, and any other magical protections Igor chooses to place on the prisoner. An arcane connection (a single hair) will be provided to the challenger. The casting of the spell must be witnessed by a mutually agreed on group of magi who confirm the rules are followed. The challenger is expected to cast his spell from somewhere far from Razhost, and is not permitted to damage the temple.
This has been attempted four times, never successfully. (If the challenge fails, the prisoner is executed by Igor.)