It's fortunate that Neon Genesis Evangelion and Warhammer 40,000 have already been mentioned, because Shinji and Warhammer 40k combines both (obviously). What Rule Of Cool-laden method will be used to kill the Angel this time? What literally impossible thing can the AT-Field be used for next? How thoroughly can Tokyo 3 be destroyed? How much CAPSLOCK can be inserted into the next In the Name of the Moon speech? What new Batman Gambit can Shinji pull off? How much More Dakka will Kensuke need? How much carnage can apocalypse Rei inflict? Which meek, unassuming and/or pathetic character becomes completely awesome due to Shinji's mere presence?
In Evangelion canon the Angels only showed up one at a time. Not in Avalon, each battle has more Angels attacking at once then the one before it. Not enough? Once the Angel War comes to an end, the team takes part in the Battle of New York against thousands of aliens, with the implication that there will be continued escalation until the Grand Finale against Gah Lak Tus.
There's this one Lemon Pokémon fanfic called The Adventures of Ryan and Kirrie which basically has an Excuse Plot covering the fact that the titular characters keep doing it every chapter. The reason it's listed here is because each time they do it, it gets more and more ridiculous, including but not limited to Shroomish spores causing them to get high (and have sex (of course) under it's influence) instead of getting poisoned or an allergy, and having a three-way with Leaf, of all people. By the latest chapter, anything resembling actual plot and Character Development simply goes down the drain and is instead replaced by what can best be described as an orgy.
If it weren't for the fact that the story's over, the Halo fanfic Enemy of My Enemy could be taken as this. The armies of the two sides are ridiculously huge (numbering several million), and the weapons get more and more advanced with each reveal. However, at the same time, as the action gets more hectic and the casualties pile on, there is actually a devolution of technology as both sides begin running out of ammunition. This does not stop the battle.
Beginning a New Path, written by stargatesg1fan1, is an epic tale of Harry Potter and the Endless Crossover with a somewhat promising opening chapter... Which quickly boils down into a seemingly endless series of bad smut and over-the-top solutions to every problem (real *and* imagined) of an equal number of fictional realities. Fourth Wall? What are you talking about? The author explicitly mentions himself as well as the cast of the movie, going so far as to have his main character merge with his dimensional counterpart and take over the production in order to make things right. Crossovers involved include the Stargate SG-1 and its spinoffs, several other Harry Potter realities, Justice League of America, AVATAR, Resident Evil, and more - and none of these would be complete without the main character adding one or more girls from the new reality to his Harem.
How long could Yu-Gi-Oh! fanfic Skin possibly get? 500,000 words? 700? 1,000,000? The story isn't even half-way through according to the author, and while YMMV, a majority of the readers are in agreement with it only getting better as it goes on in its violent path of mind-screw dreams and over the top card-battles.
The same as above can be said about Dragon Age: The Crown of Thorns. Counting almost 650,000 words, it has been a constant fount of epic gambits epic battle scenes, snarky humor, character developments and overall badassery on the part of the seven Grey Wardens, culminating in the most recent chapters with a Battle in the Center of the Mind between the Guile Hero / Magnificent Bastard dwarven noble protagonist and the Archdemon, a confrontation that ends with the latter gaining a higher degree of sentience. All in all, there is no telling what will happen next because a major plot point has been changed, and it was definitely not the only one.
Cornova and Jakayrta (writers of Poké Wars) have a habit of making each battle more epic and destructive than than the last.
The Oogakari in Yet Again might as well be the trope codifier for Fanfiction.
Uninvited Guests...well, let's say it starts from Yachiru burning entire 11th division brrack down while trying to bake a cake and currently at the point where the Gotei 13 using Komamura as the filler villain in order to re-claim the Plot Armor that Aizen hi-jacked from Ichigo. Yeah, it's that kind of fic.
Thirty Hs. From space armor to groinsaws to killing the fuck out of planets using another planet's corpse and vampire cavemen on Mars, it might as well be called Harry Potter: Serial Escalation
Imperfect Metamorphosis: How much worse can Reimu's day get? How can the situation get any worse? Exactly how much can we push Reisen to the edge of despair?
The MLP fanfic The Immortal Game embodies this. The story starts out with the Mane 6 pitted against an evil spirit, and progresses to the point where ponykind is fighting for survival against an army of monsters, and later the creator of their universe.
And while we're on My Little Pony fanfics, Fallout Equestria: Project Horizons does this all the time. How many times can the author make the backstory even more tragic? How much Nightmare Fuel can he cram into prewar projects? How many times can Rampage messily die in a chapter? What kind of balls-to-the-walls escape plan can Blackjack come up with next, and how drunk will she be while doing it?
The Swarm of War starts with a few hundred humans, a few hundred Orks, and the Overmind with a few dozen Zerglings… Ends up with human armies a million strong, hundreds of Chaos Space Marines, hundreds of Daemons. hundreds of thousands of Chaos Cultists, millions of Orks, and tens of millions of Zerg (with all kinds of advanced tech brought out). And all of that is before the expansion to other planets starts…