Thanks! In all actuality, I've only been on one dig (although I do have a degree on it), but I'm planning on going on several more. My work mainly is in epigraphy, that is, ancient inscriptions. Essentially, I'm going to either try to dig them up before other people find them or find ones that have been dug up in museums (but are unpublished) and then write on them.
However, on the one dig that I did go on, we did find a number of things. To begin, normally when you're digging you do find a lot of different animal bones. This is because people either eat the bones and dispose of them just on the ground or because animals come in later and either die or eat other animals, which then goes into the sort of dirt fill that sits above the original abandoned floor. Now, these aren't fossils per say- the original organic nature of them still remains. Hence, they can be carbon-dated (And also, if you're unsure whether something is a bone or a rock, you can lick it, and the bone will stick to your tongue. Fun disgusting fact about archaeologists!).
There were two interesting cases I had with animal bones on my dig. First, one time we did fine deer antler that was around ~2700 years old laying on someone's floor. This is significant because deer just don't normally walk into people's houses and die, but rather someone has to kill it and put it there. So, it seems like someone was using them for decoration or possibly as some sort of tool. Secondly, there was one time where we were opening a new square (that is, digging into a new segment of fresh ground) and as we were digging we picked up a jawbone. It was rather large and my first inclination was that it was something like a donkey's bust. But then the sort of harrowing realization hit me that what I was holding was probably actually a human jaw bone. And as it turns out, it was. Basically, we found an entire corpse scattered in that square in many different pieces for the next several inches as we went down. Like, there was a lot of human remains. We think the body was probably somewhere under 200 years old and a more recent burial, but I never heard if they ever did radiocarbon dating on it. Needless to say, it was an unsettling experience as we picked up the bones from this person who died and then put them together in these little brown paper baggies, which is similar to what we did with animal remains.
Also, I didn't lick those human bones to see if they were rocks or not.