Wow...so many. Most not as epic as many people but a lot of them. Where to begin?
My first truly epic kill (or in this case, kill streak) would have to be Unreal Tournament 2004. Playing the mode with the ball launcher in the big ass arena (idr either name) I scored 16 "holy shit" kills in a row (which is itself already like the 7th kill before you get those). Best part: 11 were enemy ball carriers, I was on defense. I had to impact hammer the last 3 because I was flat out of ammo!
While I'm on that, Unreal Tournament 3 cinematic-style point-blank jump kills with the Flak Cannon. More of those than I can count. I'm not usually an up-close-and-personal shooter player (never use shotguns in anything, heh) but Unreal has always been my exception. Nothing beats going eyeball-to-eyeball and filling your enemies face full of molten lead.
That...and I once took out an entire team with a remote-controlled redeemer in UT3. We were losing onclaught 2-to-1 and they decided to turtle up, sent one of 8 guys to attack random nodes and kept the rest of their team of defense, I believe it was Torlan. Anyhow, got the redeemer, sniped the one dude trying to kill our tank node, then whipped out the redeemer and wiped out their whole damn team right after he respawned. We still lost that round, though :-/
Besides Unreal...hmm...I mostly play strat games so...oh! In Supreme Commander: Forged Alliance I once completed the first mission with nothing but a single sACU and a Fatboy. No seriously. I usually turtle in RTS games so not counting a ton of base defenses I build only those 2 units. Wiped out every hostile on the map with it too - literally. I made a point of killing the Seraphim ACU dead last, after every single unit and structure was toasted. Never been more proud of any RTS strategy than that. Well...except...
In Command & Conquer 3: Kane's Wrath, I once defeated 5 Hard AIs in Skirmish as the Marked of Kane with 11 units. I used 9 Spectre Artillery (that I used the cloak power on so they were perma-cloaked) and 2 Shadow Teams. Defeated them all with called-in artillery strikes. Well...I had 1 squad of Vertigos solely for Ion Cannon elimination duty but aside from just that, only the Shadow Teams and Spectres. I was especially proud of this because Shadow Teams are SO fucking easy to kill I was shocked I only had to retrain them once.
Back on more direct games...I cleared out all the Deathclaws north of Sloan in Fallout New Vegas using only Ratslayer. Not the most spectacular feat but not the easiest thing to do, either. That playthrough I ended up in New Vegas way to fast and never tried that again - there's a reason they make you go around. The game is WAY too short and boring without that.
I guess maybe a few things in Skyrim? I sided with the Imperials every play except one, and in every play including the Stormcloak one, I always kill the final target with Unrelenting Force. It has such a ring of poetic justice to Fus Roh Dah Ulfric Stormcloak to death. Not really a badass kill as just an artful way to go about it.
Speaking of artful kills, Dishonored, in more ways than I can count. I used a springrazor and posession to take out many, MANY hostiles. Also Springrazor Bombs - attaching a Springrazor to a bottle, then pick it up and throw it so it detonates on impact, not silent but MUCH quieter than Grenades. Really, almost any kill in Dishonored automatically carries a level of badassery above average.
Borderlands 2 many times. My favorite to date has to be when I killed the Warrior on my Gunzerker in a single second with pistols. I freaking hate the Gunzerker. Least favorite class or character in any FPS I've ever played (well almost). That said, DPS is DPS. I theorycraft like nobody's business so I set one up with a special talent build (Divergent Likeness, Keep Firing, I'm Your Huckleberry, etc) and a Sheriff's Badge. Then I gave him dual 10.4 speed Infinity's. Then I just killed everything always and fast. Back before the UVH pack, he was doing 12.4 million DPS PER GUN! I haven't played him at all since but I'm willing to bet it scales exponentially, just like everything else.
And...I've got about a dozen more but I'm just gonna stop unless someone says they really wanna hear this from me, heh.