I've had the pretty common ones that have been mentioned here, bitten by my cockatiels, swooped by magpies, chased by a pelican etc.
I've also had three pretty serious encounters with animals, that a lot of people don't fully believe when I tell them, but they always make good stories to tell kids when I'm teaching.
In Thailand I worked at a refuge for a period of time, mainly with Asian Elephants. We were cleaning the habitat of a trio of elephants that didn't like people at all and had to be fed at the far end while we went in a picked up their mess (probably about 500 metres away or so.) One of the elephants snuck off when they were being fed and somehow snuck up and appeared about 20 metres in front of us all and charged. Everyone was able to escape in time, as it was probably just a warning charge, but there were a few injuries from people trying to jump through the thick wire fencing.
In went to a animal park in South Africa that I immediately regretted. It was 8 am on New Years day and I was the only visitor their, so I got my own personal tour on the safari van. In one of the paddocks was a large pride of lions, with one of the females holding back as we went past. As we went through the paddock, I heard a loud scream, then all the lions in the park begin to roar, it was a fearsome sound. I looked up and saw the lioness had a park attendant by the throat and the pride was running towards her and the attendant was quickly being pulled at by his arms and legs. A van ran out and the lions dropped him and spread, except for the lioness, who then came to the van and started to sniff at the door and pushing against it, trying to get in. Luckily the door was more secured then the gate, and we drove off, leaving the lions still roaring. Thankfully the attendant wasn't killed, not sure the offer of a free tour another day was exactly the best way help someone who had just witnessed an attack.
Lastly I was in Uganda for a gorilla trek. When visiting the family, a three year old male started jumping on us and starting to wrestle, while none of the others in the group were fussed with us. We got to see a three week old baby gorilla pretty close too which was amazing and the guides decided to takes us a bit deeper in to have a different look at it. Unfortunately we walked in between the eyeline of the baby and the massive silverback, who wasn't to happy. He stood up, beat his chest, bellow (sounding exactly like Donkey Kong in Mario Kart) and ran straight at us, shoving over the person directly in front of me. He then calmly went and sat back down where he was and continued to eat. Once my heart beat slowed down I realised what an amazing display I had just witnessed.
The elephant and gorilla charges were pretty amazing to witness, mainly after the fact, as I'm pretty sure they were both warning charges.