As everyone has indicated, starter supplies in a rental can vary, but one thing we have seldom found on arrival is a box of tissue. Someone in our party ALWAYS has a drippy nose, so I make sure to toss a couple boxes in my suitcase.
My packing strategy is to take as many lightweight, consumable products along as there is space for in my bag. It prevents things from shifting and I can do a little shopping and know that I've got some room in the suitcase. I like to wrap things in clean plastic grocery bags, as well, which provide some leak/rain protection (if you have ever watched suitcases sitting on an airport tarmack in a rainstorm, you will understand that!) These can double as trash bags. I actually have dollar store packs of cone & basket-style coffee filters which just stay in my suitcase, and usually take along a small amount of my fave coffee so I am ready for the first morning.
Fabric softener sheets are another must for us. They weigh nothing, take no space, so bringing along a week's supply from our mega-packs at home beats purchasing a small box!
Enjoy your trip...once we discovered vacation rentals (have done condos, townhouses, and larger homes) we can't ever imagine a hotel again!