It really depends for me, because different different genres on the extreme metal spectrum can be extreme in different ways. I think for me a big part of it is at the root of the progression of extreme metal and the desire to play something faster, with increasingly abrasive vocals, more distortion, deviation from standard tuning whether down-tuning or tuning higher, and an overall rawer, more primitive atmosphere.
I think another of the most important elements in it for me is the chaotic sound, which may or may not actually include an element like speed and even the rawness might be replaced with pure sonic abrasion, though I find it difficult to find anything with crystal clear/flawless production "extreme." Marduk is faster than black metal era Bathory, for example, but I will always think early Bathory fits the extremity bill more than Marduk because of that raw, heavy and chaotic sound. It really depends on the "type" of extremity. Some bands may play slower, yet their riffs and overall tone are far more foreboding and crushing than a band that plays at blistering speeds without totally mind-slaying riffs yet to my ears may sound like a bunch of pre-schoolers trying to sound "extreme" not knowing how to do it. Then again, you have a band like Vader or the Berzerker who are both very fast, great riffs (yes, even the Berzerker lol), very abrasive, down-tuned to hell, who may lack the "rawness" of other bands yet are still extreme. I guess for me as well there is a difference between a band consisting of talented guitarists and drummer who play their instruments well and technically proficient, and those bands who claim to be "technical this" and "technical that." If the focus is alone on being "technical" for the sake of it, it loses part of its extremity for me. It becomes too superficial, hence I'll often prefer bands that might sound somewhat "sloppier" despite executing great riffs and solos, to bands who do it with machine-like precision.
I guess all in all a band that is extreme and makes me go "holy fuck, the apocalypse is here and I'm fucking stoked" are bands that sound like the four horsemen are here, the sky is fucking cracking, the earth has just been fractured and the hordes of hell are spilling forth to join you in destroying civilisation, plagues and locusts are rampaging the earth, which can take the form of Morbid Tales, Under the Sign of the Black Mark, Crossing the Fiery Path, Pleasure to Kill, Illusions, INRI, De Mysteriis Dom Sathanas, A Blaze In the Northern Sky, Worship Him, In the Nightside Eclipse, Like An Everflowing Stream, Consuming Impulse, Mindloss, Realm of Chaos, Litany, Servants of Darkness, The Berzerker, Gore Metal, Scum, it doesn't fuckin really matter.
That said, there are other breeds of extreme metal that may not sound like hell is on its way, but still invoke a feeling of extremity. I'll take the most classic example of that: Take En Ring Til Aa Herske as an example. Ever since I first heard the song I got this Gollum-like vibe from the spoken vocals on it. I could imagine some madman or creature in a dark cave rambling to himself, voice echoing through the cave. No place for him among humans or anywhere else and so took refuge underground, but he doesn't remain there a pathetic shadow of a being like the real Gollum. He rises up, sheds his chains and declares that he will not serve neither mankind, nor any deity only himself. In essence extreme metal secretes this: rises above everything and exists for itself alone. That sounds like it's being emitted from a torture chamber yet it's a kind of rise to glory, and screams of torment become battlecries. I really do think vocals are huge part of what makes a band extreme, but not always. For example, melodeath often exhibits deep growls, yet I don't perceive most melodeath as extreme metal. Same with deathcore and metalcore, no matter their vocals these bands will never be "extreme" for me. They're a mimicry of extremity. In that sense, the spirit/ideals of the band is important as well. Extreme metal initially came from some form of disguntlement with mainstream music, often alongside western values and the need to rebel and to play something faster and/or more "evil" and extreme than the next guy, and to maintain certain founding ideals. Both music and ideals do not run parallel with what is expected in mainstream music. There is no attempt to appeal to those who don't "get" extreme music in it's rawest form.
To answer the second half of the OP about bands that have really struck me, a lot of them are the well-known or lesser-known classics of extreme metal as well as some more obscure bands like Morbid Scream and Nocturn. Of course there are modern albums that do this for me; the opening of Vader's Litany continues to blow my mind. Probably one of the most brain-rattling albums in modern death metal for me. I actually only ended up checking out Gorefest this summer when I came across The Eindhoven Insanity for 7 euros in a second hand shop in Germany and decided to finally check them out since I'd always heard their name mentioned but never got around to them. At first I found the vocals monotonous, but since then I've really come to adore early Gorefest and have come to think that next to van Drunen, De Koeijer may be the best vocalist to come out of Holland. When it comes to extreme metal Holland has a lot to be proud of. Extremity has made an impression on me no matter if coming from the Swedish death metal scene, Norwegian black metal scene, American thrash and death metal scenes or so on. Beyond the above I don't know what to say, and well, most of this has just been rambling on attempting to explain what I love and have always loved about extreme metal so it's time for me to shut up before it becomes more nonsensical and chaotic.
Edit: I thought I'd add, I think the old school extreme bands will always be the epitome of extremity in metal as far as the chaotic, high energy and raw elements of extreme metal are concerned. And to add to that, comparing old Hypocrisy to what they became, I think it's really that heavy, grinding guitar tone, bass and the heaviness even of the growls themselves that really start to lack after Osculum and get diminished as they went on, and I think that highlights really strikes me as great about extreme metal. Taking what is great about metal already and turning it up tenfold.
And as I finish this post Hypocrisy's cover of Black Metal has come on and I feel it's totally fucking fitting to end this long ramble. So on that note, I end this post with "Lay down your soul to the gods of rock n' roll! Metal ten fold through that deadly black hole! Riding hell's stallions bareback and free, taking our chances with raw energy!" and a final "uuuugggh uuuuugghhh black fucking...METAL!"
