Well, QL, there are the obvious ones.
Combat matches and jousting with non-lethal or less than lethal weapons.
Then you have ancient forms of competitions:
Footraces
Horse races and mounted races
Race courses with flags and turns
Javelin throws for distance and accuracy
Discus throws for distance and accuracy
Hammer of Weight Throws for distance and accuracy
Wrestling and boxing matches
Then you could have more modern forms of such competitions:
Obstacles courses
An Iron Man or Triathlon competition
Strong man competitions such as lifting heavy stones, bending iron, log splitting, tugs of war, animal pulls (see if a team of men can pull a horse)
Swimming (distance and speed)
Hastiludes such as knife and axe throwing, dart throwing (for distance and accuracy), titling at the target (whatever that is), fights (wrestling matches) against muzzled dogs or wolves, or against small bears, etc.
You could also interweave other types of competitions which are not strictly combat oriented, such as poetry and singing and rhyming competitions, magical competitions, stage magic and illusion, bardic recitals, mnemonics competitions, complex tapestry or design making, animal husbandry and control, metallurgical skills, craftsmanship of various kinds, artistic displays and crafts, etc...
You might even think about cutting down trees and building small defenses (two or more teams of men quickly cut down trees and fashion usable and suitable defenses, as the Romans used to do), weapons and bow making (people have to make their own functioning weapons), tracking and hunting competitions, climbing matches, and of course survival courses and obstacles.
If you're gonna include things like Rogues then you might consider acrobatics, athletics, games, team sports or maybe even things like pickpocketing competitions.
If you're gonna include things like Clerics and Monks then you could include everything from displays of power to blessing contestants to curing disease to public debates.
And don't forget drinking and feasting competitions. As well as endowments to the poor and the King's blessing.
Those go over big.