We ran across a rattle snake slithering across a cache on a pile of rocks. Later that same week I was reaching for a cache and I noticed just in the nick of time that a big fat black widow spider was right in front of it. Almost stuck my hand right into it. I kind of got the feeling that nature didn't want us geocaching that week.
Most annoying animal we have come across is ticks...millions of ticks. Even though I don't think they count as an animal.