Here's something I've pondered for quite some time actually, but never bothered to look into in the past. My question is simple: what's the point of having a larger than 100% percentage on a skill? Now, I understand that with guns, the higher percentage in a weapon skill increases your chance to hit from greater distances, sneaking members don't get detected as easily and maybe a doctor can heal more HP and fail less often when the percentage is as high as possible. But what about skills like unarmed and melee weapons? The chance to hit never exceeds 95%, and that can be easily achieved even with less than 100% know-how in that skill. Do they start to do more damage if the number is, say, 120% ? Is there any point putting skill points in unarmed or melee after a certain point?

I don't think more ranks in the Doctor skill will let you heal more health per try.

Higher skill in Melee and Unarmed skill is useful when you want to make aimed hits. Like hitting somebody between the eyes for example.

While Doctor and First Aid can cap nicely at 100%, other skills benefit greatly from exceeding 100.
Barter, in effect, doesn't offer you equivalent value at 100% proficiency. The truly obsessed trader will need very high Barter proficiency and, in all likelihood, an additional Perk to improve prices.
Melee, Unarmed, or any other combat skill at 100% will offer you the maximum possible chance to hit only if your opponent has used all his AP, is not wearing armor, and has no natural armor class from agility. Believe it or not, this means you won't have a maximum-possible hit chance against deactivated robots. (Note: Some armors mitigate damage well but deflect it poorly.) In order to reliably strike an armored opponent, you may need 130% by the late game. In order to ignore armor reliably, you may need at least 160%, but that assumed called shots, such as to the eyes. Simply hitting high level opponents is sometimes a matter of respectable skill.
Many non-combat plot solutions are available only with very high Science and/or Speech skills. By 'very high' I mean above 100%.
Other examples exist. I should try to figure out which is which and make something of an inventory, but at present I'm quite busy. I hope this helps.

While I made the thread I totally forgot about new unarmed strikes becoming available at higher skill number though. Like piercing strike for example.

By the way, funny thing: I'm playing Fallout Tactics at the moment and managed to train Target, the tribal, into a master nurse approximately by the time I got to Junction City. He has 222% First aid (gave all my skill books on the subject for him to read) and at some point I actually got a little surprised about how high the skill had actually gotten. It can be surely said though that having that skill percentage so ridiculously high doesn't make him that much better at healing though; using a paramedic bag often heals only about 70 hp, although it does fully heal often too.