I had to have been a cat of some sort. I've sat for several people's cats, and rescued my share of them too. They're my absolute favorite kind of animal (all kinds, not just domestics).
At one place I was able to pet 4 of the 7 I was told I wouldn't likely be able to at one house (plus the 6 that aren't as shy. . . the owner was unable to give up a foster litter, then has subsequently rescued a couple of feral litters

). When I went to meet the cats beforehand, one of the 6 came right out to check me out, but she apparently won't come out for the owner's mom (who the cat knows) for at least an hour. She was guarding me in bed at least one of the nights.
At another there is a shy cat who hid the whole first day I was there, but now she comes out within half an hour of my arrival (though she still doesn't eat like she should, at least she eats now) and I keep being told the other cat is also shy, but she really isn't around me.
At the third place the woman has an ancient dog, but she also cares for 5 ferals (one has gotten close enough to her to spend time inside the house, except when I'm there). Two of them began letting me pet them at meals during the third visit (they go away a lot). They don't really even like her husband. Sometimes one of their daughters goes with them wherever and her cats come over to stay too. I was left a note (as usual, the dog is really picky), and I had just gotten to the part where it was telling me how shy they are and one of them is on the counter in my face saying hi. The other came right out to have a look at me. I let him come to me when he was ready, which was less than an hour after I got there.
I'm set to sit for another lady in a couple months, but she wanted me to come over quite early to make sure I'd be ok with the job in case she needed to make other arrangements. She has a couple of pretty friendly cats and one that has had a pretty tough life. The latter one ran and hid when he heard me arrive (as is his custom), but they were figuring they were going to have to catch him to show me how to groom him (he's a maine coon). He walked right over to me when we went on the porch where he was, sniffed my hand, then tried to ask to be let in.