In the USA on the PCT it happened a few times that I heard a bear just metres away from my tent. They were black bears that are after things with an interesting smell and are not likely to attack you, but you know they can kill you so it is definitely a bit disconcerting.
Another time, late afternoon (still daylight), there was a stampede of sorts of ten (ish) small deer that were flying for something and they ran within a metre past my tent. I was really afraid they would hit my tent. Later I discovered that I was camped on top of a faint animal trail, which may have been their usual route, that now had an unexpected tent on it.
Last November in Australia, there was a nocturnal bird that dive-bombed my tent twice (and pounding the roof). Just before it touched the roof I could hear it approaching and could see the fabric moving from the air pressure of it's wingbeats. I wasn't afraid (what can he possibly do?) but is pretty disconcerting when you are in the pitch black in the middle of nowhere and don't know what exactly is happening.
I also heard kangaroos hopping around my tent at night and sometimes I was afraid they would jump on my tent by accident, because normally there wouldn't be a tent on that spot.
However, the most scary things are always other humans. I always try to camp (far) away from roads, but one time in Washington state in the middle of nowhere I struggled to find a spot (too lush vegetation). Finally I found an open spot where I pitched my tent. However, though in the middle of nowhere, there was a 4x4 track (about 100 metres from my tent) and later that night a car appeared there and stopped for a while. Why?? I see no reason. After a while he moved on, fortunately. I don't know if he even spotted me, but I found it a bit scary. It seemed like strange behaviour in a place where I wouldn't even expect a car in the daytime, let alone at night. If it is some deranged person with a knife of a gun... On the other hand, it may have been a just hunter trying to spot deer, because later that night lots of deer arrived there to graze. And they were extremely skittish, because I heard them immediately run off when I switched on a tiny light in my tent, which may indicate that they get hunted there.