I looked up to see if I could see the towels - nope. So I went upstairs to see if I could push them down. I stuck my arm as far down as it would go and didn't feel anything. So I did the same thing on the second level - same thing. Nothing there. I threw a couple more hand towels down from the 1st floor and those made it through so I have now confirmed that the clog is somewhere between the 1st and 2nd floor. I figured that maybe throwing something with some weight might give the towels enough of a push to make it through, so I decided a hammer might just do the trick.
Alas - it didn't work. I now have a bunch of towels + a hammer stuck in my laundry chute. Hopefully my husband can fix everything when he gets home.
Lesson learned: Don't shove everything in the laundry chute at once. It really isn't that big.