As the whimsical responses suggest, it might be best to confine this to actual cases of elephants being killed.
It has already been noted that humans were capable of this well before they had guns. In Man-Eaters of Kumaon, Jim Corbett relates a case in which two tigers killed an indian elephant. He makes it clear that this was most unusual.
Another elephant?
Not an anaconda but the reticulated python of southeast Asia is on record as being the longest at 10 meters (32 ft 9.5 in) for a specimen shot in Celebes, Indonesia, in 1912.
What about a pissed off and determined woodpecker?
In Ice Age the whole pack of saber-tooth tigers was afraid to take on the mammoth. I rest my case. 
I once killed an elephant in my pajamas…
Ok, I’ll bite. What was it doing in your pajamas?
Probably a number of parasitic insects could kill an Elephant, under the right circumstances.
And there are Elephants who regularly swim between ocean islands (islands that aren’t very far apart, mind you), so it’s at least theoretically possible for them to be attacked by large marine predators capable of killing them. (Even if it’s not very freakin’ likely.)
Say, do human ancestor and “cousin” hominids species’ count as “animals”? Some of those guys would probably have been able to kill an Elephant. (Some of them probably did, too.)
And of course…there are many Dinosaurs that could probably kill an Elephant with relative ease.
Maybe a scorpion?
Or even a mosquito - do elephants get malaria?
1st because I don’t know…are you male or female. If you’re a guy I’d say yeah right. You wish you had a trunk down there…me too for that matter. A prehensile penis, that’d be real handy. :eek:
If you are female, I’d say lucky girl.
You beat it to death I guess. 
After thinking about it for a second I realize you must be female 'cause I don’t know any guys that actually wear PJ’s. We get for gifts from our moms and grandma’s. But we don’t ever wear them.
I’ve got a pair that are so old they’ve got feet them. heh-heh-heh… 
This brings up another question. Would a King Cobra even mess with an elephant?
I’m no herpetologist, in fact my experience with reptiles is highly limited and I know nothin gabout them.
But I’ve always just assumed/thought I heard that snakes only kill things that they think they can digest, I find it highly doubtful any snake in the world could swallow a full grown bull elephant.
As for hippos IIRC hippos cause a huge number of deaths because they accidentally knock over canoes and other small boats. Once the boat is tipped over the thrashing people upset the hippo who can become threatened very easily and it will in turn either start thrashing, or attacking, or anything really. Just about any rapid movement from those monsters can cause people to drown.
There are no known natural predators of elephants other than man. They are much more likely to die of a heart attack or a parasite than they are to die by extra-species violence.
An anaconda or python will only attack what it can swallow. I doubt it could swallow more than one of an elephant’s legs. Even the truly monstrous specimens (longest documented is about 8.5m) could probably not deal with more than a large pig or a small cow.
Like I said, there are no predators that hunt elephants except man. Since elephants aren’t carnivores and a snake could never swallow an elephant they aren’t naturally in competeition with each other.
The OP merely asked, "Is there any animal capable of killing/seriously injuring a full-grown elephant?"
In response I’d say YES…capable however unlikely it may be, the ability is there.
Seriously injuring definitely!
I recall reading somewhere about a bison or a buffalo killing an elephant. Does that sound familiar to anyone?
I’m guessing no. Because elephants aren’t made out of wood.
Not in the sense of deliberately taking one on, no. But in my experience, snakes will strike defensively if they are accidentally cornered and get scared. I suppose that could theoretically happen through an accident of terrain or something.
The King Cobra doesn’t even make the top ten.
Five of the top ten could be found in the bushland around where I grew up. A father of a friend of mine was amongst the first to identify No. 1 on the list. I can recall at least two occasions when we found specimens of No. 2 on the list in our backyard and I still remember being hurriedly rushed indoors (along with the rest of the class) when one was seen in the playground of my kindergarten.
Which is no doubt why we never saw any elephants in the bushland where I grew up.
We were on a safari in Kenya a couple of years ago. We drove past an elephant skull and some bones. Our guide (a local Maasai warrior) claimed that the elephant was brought down by lions a couple of years before.
Of course, I would imagine that lying to tourists is like a national past time for some of these folks, but there you have it.
A Ninja.
Non- extinct animals only? Dang.
Certainly a pod of orcas takes on targets larger than elephants, but it is probably hard to arrange a fair steel-cage match between them.
Salt water crocodiles get pretty freaking big (6 m or more, 1000kg). While certainly much smaller than the elephant I can see one (desparately hungry) ambushing an elephant crossing water a little too deep and doing some serious damage.
http://www.angelfire.com/mo2/animals1/crocodile/asc.html
Trick is even if there were an animal or animals capable of the job (pride of lions or whatnot) it would be pretty dang hard to persuade them to attack - the payoff is too small compared to the risks.
And incidentally, therefore, are not witches.
You can still build a bridge out of them though.