In the anime Gunsmith Cats, "Minnie" May Hopkins has a slight tendency to use home-made concussion grenades in inappropriate situations. However, she's every bit as skilled in their use as Rally is with handguns. A single one will just leave a couple of gang members stunned for a minute or so. She can use three to blow out a pursuing car's drive shaft. And when she's really pissed off, she'll set off a dozen of them at once, causing extensive damage to whatever building she's in (typically shown with a Discretion Shot of the outside of the building with all the windows breaking and billowing out smoke).
In Rurouni Kenshin, Sanosuke's former resistance comrade Katsuhiro Tsukioka employs this tactic, with the aid of flint rings on his fingers to ignite the fuses as and when needed.
Henya from the Juppongatana uses dynamite sticks to both sustain his high-flying style and attack his opponents.
In Katekyo Hitman Reborn! Gokudera is a bomber with the personality to match. Although for firey personality he displays good control over his projectiles.
Isis in Magical Record Lyrical Nanoha Force battles through the use of various explosives and other chemicals. This allows her to go toe-to-toe with the villains who have been all but invincible against the previous heroes, as their Anti-Magic doesn't help against getting blown up.
Jillas in Slayers, a [Gunslinger] Fox with a penchant for using any sort of explosive means.
Minene Uryu from Mirai Nikki, completely in sync with her reputation as terrorist.
Lan Fan and Fu from Fullmetal Alchemist frequent the use of bombs in addition to hand-to-hand combat. While Fu uses various types of explosives such as flash bombs and tear gas bombs, Lan Fan almost exclusively uses grenades. Edward is able to confirm her identity after being caught in one of her explosions, (recognizing the previous destruction she had caused, which nearly killed him).
There is a grenade-throwing IRA mercenary in the Sin City story Big Fat Kill. He mentions that while he prefers using grenades, he'll use his knife against Dwight since he killed his friends. At that point, It's Personal.
Spider-Man's arch enemy, the Green Goblin and those who followed his designs all use pumpkin-shaped grenades as their weapon of choice. Though they will use explosives not shaped like pumpkins if necessary.
Bomb Queen fights with powered gauntlets that let her toss out miniature sticky bombs.
Malazan Book of the Fallen has Morath munitions which are primitive explosives contained in ceramic jars. Malazan armies use them with devastating results to the enemy. They are carried by sapper specialists who are usually elite soldiers. The regular troops consider sappers to be madmen for their willingness to carry and use the explosives.
Grenedier, one of the villains from the Kingdom of Champions supplement.
Blood Bowl's goblin star player Bomber Dribblesnot.
In TimeSplitters: Future Perfect, enemies can only carry one weapon at a time and grenades are treated as a specific weapon (compared to 2, which had some weapons with underslung launchers and various minesnote Which were essentially Sticky Bombs that came in timed, remote control and mine varieties. but otherwise lacked them, probably because explosions were always an instant kill). Meaning any equipped with grenades will only use grenades. This is lampshaded in story mode;
Everyone in Team Fortress Classic has some sort of grenade tailored to their class, and gameplay styles were generated from using them. In Team Fortress 2, only the Demoman lobs grenades around.
Multiple characters in Bomberman, including the eponymous character, as one might guess from the title.
In Battle Bugs Ant, Bee, and Pill Bug throw bombs. Robber Fly can use a bomb too, but only after stealing it from enemy bugs.
Various bomb and grenade-related powers can be obtained in Freedom Force. Liberty Lad in particular specializes in them.
Characters specializing in the Throwing skill in Jagged Alliance 2 can be trained as one of these after they're given a proper supply of grenades. Very useful on night maps: grenades don't give away your position, for one, and the same skill applies to throwing knives, which can One-Hit Kill unaware enemies.
X-COM soldiers can specialize in grenades, but the game's limit on items that can be carried to the battlefield makes it rather impractical. Aliens make very powerful grenades, and sometimes carry no other weapons.
Bug!! had the snail boss... which grew a helicopter out of its shell, flew, and started dropping increasingly deadly bombs all over the arena. There were also literal army ants that shot out grenades from their tails.
All games also provide Snake with usable grenades and even a grenade launcher at times.
In Super Smash Bros., Snake has a variety of explosives at his disposal. Due to realistic bullets being too small to see in regular gameplay, Nintendo opted to have Snake use explosive weapons (which are rarer) in Brawl. He can use a mortar launcher, C4 packs, frag grenades, claymores, remote control missile launchers, a RPG and, for his Final Smash, a grenade launcher.
From Panzer Bandit, playable character Ein can throw bombs two-at-a-time or roll a single one towards his target. His Mirror Boss Tsuvai can only throw two bombs in a close mortar-like angle.
Big Bad Dr. Farado has a Spider Tank whose best move is to vomit a dozen bombs into the enemy.
In the Tenchu series, bombs are usually among the selectable items to carry into a mission. They pack a very mean punch, but are usually not as effective due to the difficult to handle and slightly delay if it miss, not to mention the high risk of getting caught in the blast radius, which really hurts for poor Rikimaru/Ayame/etc.
In terms of characters using bombs, Fatal Shadows has Hitoha, who uses incendiary bombs along his fire gauntlets.
From the classic Mega Man series there are several Robot Master, like Bomb Man (1), Crash Man (Mega Man 2), Napalm Man (5), Burst man (7), Grenade Man (Mega Man 8) and Commando Man (10).
The Engineer profession in Guild Wars 2 can equip a backpack full of assorted types of grenades, throwing betweeen two and six with every use.
The Knight class in King Arthurs Gold can carry up to three bombs, which explode in Bomberman fashion and destroy both blocks and enemy units. Interestingly, the thrower can be hit by the explosion (to facilitate bomb-jumping) but not other teammates.
League of Legends has Ziggs, the Hexplosives Expert. He fits this trope to a T - he carries such a huge amount of bombs that, being a yordle, a member of a teensy furry race, he is more bomb than animal (by mass). He throws them as his basic attack and two of his abilities, one of which produces a whole cluster of them. His ability repertoire is closed by a remotely detonate-able satchel charge and a Mega Inferno Bomb, which is powerful enough to produce a mushroom cloud.
In Fallout: New Vegas, one can encounter the Powder Gangers, a gang of escaped convicts who favor dynamite and are not afraid to throw it around liberally and in large quantities. Then there's the Boomers, who go from merely favoring bombing everything that opposes them to building a sort of cult of explosives. And it is totally possible to create an explosives expert, throwing dynamite and grenades, then moving to grenade launchers and even a fat man (Portable miniature nuke launcher), with the added benefit of being able to plant/disarm mines more effectively.
In Chack'n Pop, Chack'n (who happens to be a cutesy Waddling Head) attacks by tossing smoke bombs to either side.
In Borderlands, Roland, the soldier, has a skill which allows him to regenerate his grenades as he kills more enemies with grenades, ensuring a constant supply of explosions. The variety of Trick Bomb upgrades available to grenades makes this a useful skill for people who can place explosives. Conversely, Brick, the, uh, Brick of the game, simply gets skills and gear that upgrades the power of his grenades. Though his grenades won't regenerate like Roland's can, they can do much more damage.
In Borderlands 2, Axton the Commando has the Grenadier skill, which increases the number of grenades he can carry. Grenadier class mods give boosts to grenade damage and put extra points into the Grenadier skill. A Grenadier-built Axton is thus, duh, very much encouraged to use grenades all the time. He also has Do or Die, which allows him to throw grenades in Fight for Your Life mode, and various skills that increase explosive damage. Krieg the Psycho uses sticks of dynamite in Fight for Your Life if he has the Light the Fuse skill, and a few skills of his own that boost explosive damage.
Dark Souls: There are firebomb-throwing skeletons in Undead Burg, and they can be quite the Goddamned Bats. On the other hand, throwing firebombs like a maniac is a valid way to fight your way through this Nintendo Hard game, especially if you're too weak to melee and doesn't have a spell to sling.
The entire premise of the game Silent Bomber is that the player is equipped with an "E-Unit" which allows him to place or throw around bombs.
In Dawn of War II: Retribution, the Ork Kommando hero, Spookums, is armed with a shotgun, but nearly all of his most powerful abilities involve explosives. He can throw Stikkbombs, smoke bombs, anti-tank Melta Bombs, and truck tyres...
In World of Warcraft, the goblin race lives by the motto "When in doubt... blow it up. If at first you don't succeed: blow it up again". If you chose the engineer profession you can opt for either Gnomish or Goblin specialization. The latter allows you to create powerful explosives that are bound to cause you some damage as well.
In Avatar: The Last Airbender, there's a group of repeated antagonists (possibly qualifying as a Quirky Miniboss Squad) known as 'The Rough Rhinos'. Each of them has specialized in a different weapon - including The Faceless, the only one of them wearing a full helmet, who specializes in explosives - specifically, he throws dynamite-sticks at anything that moves.
On the hero's side, there's Chey, a member of Jeong Jeong's group who uses small grenade-like exploding balls.
Batman Beyond gives us Mad Stan. His catch phrase (and approximately half his spoken dialog) is "BLOW IT ALL UP!" Yes, the bold and caps are necessary.
Averted in the Grenadier regiments of 18th-century European Warfare. Grenadiers had to be the biggest, brawniest, and most disciplined of an army's troops. Usually tall farmboys, they had to have the strength and force of will to stay in a neat line and march, in-step, to within twenty yards of the enemy (the effective range of a musket being 80 yards). The survivors would then throw 'very' temperamental lumps of gunpowder and iron the size of their own fists (or bigger) at them, and fire their muskets before charging, screaming wildly to channel their fear, into the probably-mostly-dead-and-dying enemy with their bayonets fixed to try and stab them to death. On the plus side, the fact grenadier units tended to have a higher turnover (mortality) rate than other units meant that most armies could afford to pay their grenadiers more than their other soldiers (dead men don't get paid), in addition to giving them extra-fancy uniforms and really nice hats.