Nice Hat/Tabletop Games

In certain section of Exalted fandom, the Dawn Caste, two-fisted fighter Demetheus is known as the hat man, simply due to a wide-brimmed hat he's been illustrated with a few times.
In L5R, Isawa Tadaka has a nice hat. Seen here. Also fairly excellent on the Experienced 2 card.
The High Elves of Warhammer Fantasy and the Eldar of Warhammer 40000 both have an irrational fascination with really, really tall helmets. The converted Avatar at the bottom left of this page takes the cake, however.
Murlynd's Hat. In the Greyhawk D&D setting, Murlynd of The Company of Seven's leather chapeaux can have near any non-magical object be pulled out of it, and it grants a degree of protection. He also had revolvers and a whip, making him probably the most badass wizard/paladin in existence. But the hat man...the hat.
The Commissars of the Imperium from Warhammer 40000. They aren't nice to anyone, least of all their own troops, but the hats are pretty cool.
- Though at least one commissar would rather wear a helmet...
- All Orks of the Blood Axe clan seem to love Commissar's hats, or similar peaked caps looted from other officers.
- While we're talking about orks, several Freebooters wear pirate hats, the most notable of them being Kaptin Badrukk's.
- It's a long running joke (with some truth to it) among 40k fans that the person with the biggest hat in any given place is the most important person there.
Warhammer Fantasy: The Empire. Every bloody soldier has some sort of outrageous wide-brimmed plumed construction, or a massive plume on a metal helmet. Except for the archers. Grand prize has to go to the triple-feathered hat in the Empire General kit, though (the one on the mounted guy in the picture).
The Chaos Dwarves in Warhammer apparently had hats whose height reflected the wearer's status. The superstructure of their warships also tended to look like giant versions of their hats.
In Warhammer Fantasy Roleplay, your character gets an 'Outrageous Hat' as a trapping if you have Charlatan as a career.
Warhammer also has the Witch Hunters's Hat of Doom. Characters have take a will test against fear just by seeing it.
- The Lizardmen Skink Hero riding a Stegadon gets a choice between a skull hat (for a wizard) and a rather badass gold stegadon mask for a fighty hero. And for all your giant mummified frog fashion needs, there's Lord Kroak's deathmask. Of course, since so few Lizardmen wear any clothing at all, they have to make the headgear badass.
The Magic Burner for Burning Wheel covers wizard characters, who as part of the traditional Robe and Wizard Hat can find themselves with the "Impressive Hat" described as follows:

This character's hat is mightily impressive. It can be seen for miles! It's so impressive that people often remember the hat more than the person.

Munchkin has several, such as the Badass Bandanna, the Helm of Peripheral Vision, the Horny Helmet, and the Helm of Courage (which is on backwards). For the grand prize, we have...the Magnificent Hat, complete with what appears to be a Jägermonster looking unusually proud. Does any other game have a hat so nice it reflects curses? Nope.
Many characters in Privateer Press's Warmachine tabletop game have pretty nice hats, but special props go to Protector of Menoth for their multitude of clergy hats and Deneghra for hers. Since the setting is Steampunk, it's kind of required to have one.
In the party game Werewolf/Mafia, a role with a Nice Hat, Big Hat or White Hat will always be revealed upon elimination, even if roles would normally be hidden. A Black Hat, by contrast, will never be revealed regardless of circumstances.
Alahazra the oracle sports a truly epic chapeau. And Imrijka the inquisitor has her wide-brimmed red number.
Magic: The Gathering has many of them, ranging from Kaldra's awesome helmet to whatever this is Voidmage Husher is wearing, but worthy of note is Innistrad, with plenty of hats handed out to its struggling human populace. To quote creative director Doug Beyer: "And hats. If we were going to a gothic horror setting then BY HOLY AVACYN ON HIGH THERE WOULD BE HAAAATS."