Rise of the Tomb Raider

As the in-game description says, Extreme Survivor difficulty is the same as Survivor (enemies are tougher and you can only get health back by using healing items and resting at campfires) but you can only save your progress at campfires.

Having now beaten the game on both Survivor and Extreme Survivor, I can safely say that this achievement is nowhere near as hard as it sounds. If you can beat Survivor, you can beat Extreme Survivor. As such, this guide will be written mainly for those who have beaten the game on Survivor. Do NOT attempt this as a first playthrough. This does stack with all the other "beat the game" achievements (thanks to Lazyslothboy001 for confirming this), but it's just not worth it.

First of all, anything and everything you learned in your Survivor playthrough still applies:
*Stealth past as many groups of enemies as possible
*Use poison arrows if you're in a jam, such as on the enemies with shields, multiple enemies standing next to each other (either having a conversation or funneling in through a door), or the annoying enemies that show up towards the end of the game (speaking of which, go for the archers first, as they're the only ones who can hit you at range, then shoot or melee the others to save ammo, as they can't hit you fast enough to be a threat)
*Always be on the lookout for healing supplies (or supplies of any kind)

For this playthrough, I'd recommend this:
*Save every chance you get. If you save and then kill a group of enemies nearby, save again. If you save and pick up a resource immediately after, save again. If you know you're about to come to a hard fight, run back (if you can) and save. It might be tedious, but it may end up saving you a lot of heartache.
*If there's a section you're having trouble with, practice in the chapter replay section. That goes double for the mandatory tombs (you don't want to do the majority of a puzzle, fall off and die, and have to start from the beginning)
*Choose your upgrades wisely! Especially early on when you're more vulnerable, having the right upgrades can be crucial (though by the end, you'll likely not need any more).

A lot of this comes down to personal preference, but here are my recommendations for the skills:
Tier 1 Skills:
Thick-Skinned - You take less damage from gunfire and melee attacks. Speaks for itself. Get this one either first or second.
Avid Learner - Gives you more XP from getting collectibles (more on that later). Get this one first or second.
Iron Hide - Lets you take more damage from explosions and fire. Also very useful. Get this one after thick-skinned.
Scavenger - Gives you more ammo from looted enemies. Resources can be pretty slim, so this one is also pretty useful.
Lightfoot - Lets you take zero damage from (non-lethal) falls by hitting to roll. Speeds things up slightly and means you waste fewer healing items.
Tier 2 Skills:
Naturalist and Survivalist - Gives you more stuff from natural and man-made resources respectively. Again, resources can be pretty slim, so these WILL come in handy.
Death From Above - Lets you insta-kill enemies by hitting when above them. Not crucial, but there are a few spots where it can come in handy (such as one spot at the end of the game).
Tier 3 Skills:
Heart of Stone - You take even less damage from gunfire and melee attacks. Again, self explanatory. Get this as soon as you can unlock tier 3 skills.
Death Cloud Arrows - Increases the size of the gas cloud that poison arrows make. This makes the already over-powered poison arrows even more powerful, so it's very helpful.
Cluster Bomb Arrows - Gives grenade arrows a secondary explosion after impact. This makes the grenade arrows almost as powerful as the poison arrows (though they're useless in stealth). I found myself running low on mushrooms a lot, so this is a good backup.
Strong Medicine - You take less damage for a short while after healing. Not crucial, but can be useful in a tough fight.
Tomb Skills (extra skills you get from completing optional tombs):
Inner Strength (Voice of God tomb) - fully heals you from zero health once per fight. This essentially gives you a free healing item once per fight (sometimes twice depending on what the game considers a new fight) and it saved my ass so many times it's not even funny.
(here's a list of the other tomb skills if you want to look, as they're not listed anywhere in the game itself: http://www.gameskinny.com/h29yz/rise-of-the-tomb-raider-9-hi...)

It's worth noting that, while this shouldn't be a collectible run, picking up collectibles and doing challenges will give you a lot of experience (much more than killing enemies) so it's worth going out of your way to get some of them early on (though you won't need to by the end). I got every collectible in Syria (the first area) when I first played it, so by the time I got to the first campsite, I already had two skill points ready to go. This is why I advocate so hard for Avid learner, as it will make getting new skills much faster.

While I can't confirm this, it seems like some resources may spawn less frequently on Extreme than normal Survivor. I definitely noticed some resources (such as the half-eaten deer in the Soviet Outpost) not coming back after I collected it once, but I think certain animals may also spawn slower on Extreme. Not a game-changer, but worth keeping in mind.

Additionally, I'd recommend saving up for the first upgraded resource pouch and getting it very quickly. This will let you keep more herbs and mushrooms with you so you can make more healing items and poison arrows respectively. You may also want to get the large arrow pouch to hold more poison arrows. Last thing, if you have the season pass costumes, I HIGHLY recommend using the Spirit Weaver outfit (part of the Remnant Pack). This will give you an almost 50/50 shot of not using either the herb or the cloth (or both) when healing, so it will save you a bunch of resources.

After doing all of this, there was only one part of the game where I really struggled, and that was the ending (after the "point of no return" message), so I'm putting up a spoiler tag for this part:


Like I said, this achievement is nowhere near as hard as it sounds, it's just nerve-racking at times (I often found myself really worried that I'd get through a long section only to die in a stupid way, but I never did). If I can do it, so can you. Aside from that final section, I only died around eight or nine times, most of which were from me missing a jump or doing something stupid. Be smart, have a little faith, and the achievement is yours. Happy Hunting.

Sorry for writing a novel for this, I'm just trying to be thorough. If I missed something or you have something to add, leave a comment and I'll edit. Do that if you leave a negative vote as well, as not doing so doesn't help anyone.