Alpine trad: What do you carry to ascend rope in emergency?

Rappel autoblock + the cord that secures the backpack to the belay loop.

I've ascended ropes with any sling (shoulder length, double length, etc) or cord (5-7 mm) that is carried on my harness during a normal climb. Really anything works. Nylon runners are great from alpine draws. If you're at all worried, add backup knots in the rope as you go higher, every 10-15 ft clipped to your belay loop. Many years ago I practiced ascending some tall trees with random pieces of gear; it all works about the same just tied as a prussik.

Well I just used two thin, shoulder length dyneema slings to ascend a fixed rope when my friend forgot their ascenders at home. Tied as a prusik It worked for me so I guess YMMV.

You can easily use slings with a klemheist knot in place of prusiks, or a GriGri in combination with another sling.

I'm surprised no one mentioned using an ATC guide in guide mode on your harness in conjunction with a kleimheist made from one of your slings used in an alpine draw to ascend the rope. Granted this isn't ideal when on super steep terrain but if your goal is super light and only planning on using this as a backup, I see no problem.

Hollow Block and Tibloc

Like others have stated you shouldn't need to be carrying anything extra in case of an emergency, you just need to learn to use the things you should already have.

It should be standard to use a full strength cord or sling for your chalk bag for any multi pitch climbing not just alpine climbing.  I used to use a piece of 5mm cord but have updated that to a double length 8mm dyneema sling.  This is a bit heavier but is twice as long and this is all you need to make a footloop/klemheist setup.  Your piece of 5mm would have to be really long for it to work in that fashion without an additional sling.  A sewn hollowblock will also work just fine but I prefer to just use a short tied piece of 5mm to make a prusik.  The advantage of having a tied piece is you can tie the prusik directly to you incase you don't have any biners left.  Also I found having a tied prusik loop is easier to adjust when using it as rappel backup (which you should be doing most of the time).  And if you need to leave the loop to backup some shitty webbing at an anchor it is really easy to do with a tied piece.

I'd recommend against bringing something like a tibloc, unless you specifically plan on ascending a rope.  I don't know a single serious alpine climber that ever brings one.  If you really want a dedicated device I'd recommend a micro-traxion.  You can do way more things with this device.  They are great for simul climbing and belaying up the follower.  These devices have become a standard piece of the rack for everyone I climb with. 

The most important piece to any ascending or rescue kit is the practice that you have put in.  Learn the many different knots and techniques out there and know how to use them all.