That sucks, sorry to hear about this. Hopefully It will grow back Like some people have mentioned. I lost about that much off of a finger and a toe, two separate incidents, but they did both grow back. They both happened when I was in My 20's or younger though, so I don't know how much age can play into It growing back.
Getting a finger caught in a sprocket looks like It hurts like hell though. Mine were both grinded off, so was a little different. The toe I was riding a moped That me a friend got running to sell. I was test riding It barefooted on a rocky asphalt road, and for some stupid reason was peddling while It was running, and My foot got wedged between the road and the peddle, grinded My big toe down right through the nail like that. Another time I was using a large hand grinder without a guard, and a super coarse grinding disc, when I was gripping the handle I ended up touching a finger to It, and It zipped that right down also. Both incidents went at least half way through the nail or more, and It took quite a bit of time, but they did grow back eventually.
I have a customer and buddy, that was working on a motorcycle and got his hand caught in a sprocket. He didn't fair as well though, and lost a couple of fingers completely. Years after He lost more on a table saw. Surprising but he actually plays pool pretty well considering he doesn't have as much to bridge with as most of us.
Long hair sleeves and all that stuff is bad around lathes as already mentioned. I cut My long hair off about ten years ago. I had hair half way down My back for many years, then realized a few things, not only with working around machines, but It's a pain in the ass to keep washed all the time, It's hot down here where I live, and God forbid, If You ever get into a scuffle It's easy to control someone's head using their hair as a handle for leverage.
The gloves aren't good around a lathe either. I was a little clouded one night working on the lathe, had a brain fart, and put some latex gloves on to keep My hands from getting messy. Luckily I didn't dislocate MY finger, but that latex really sticks quickly to a work piece, and I came close to It. I pulled My hand out just as It was starting to wrap MY finger, the latex snapped and I got My finger out in the nick of time. I'll never wear any kind of gloves on a running lathe again.
Anyway Mine grew back as I mentioned, but they were grounded down somewhat cleanly, and I never even went to the ER. That sprocket looked Liked It chewed It up pretty good, so I imagine It could be a little different situation.
Here's hoping It heals quickly, the pain in minimal, and does not last long.
Greg