US Currency is printed on a special paper made by Crane Company. It's got a lot of cotton in it, so much so it's almost a fabric. I believe it's got a special kind of denim in it as well, but it's been a while since I researched the subject.
It's quite washable, although repeated washings might eventually deteriorate it. It is recycled by the Mint when banks return deteriorated notes for new ones. Can't remember what it goes into, probably just general paper pulping.
Which reminds me. The US Treasury has been making noise about retiring the $1 bill for years now. They've come out with a series of $1 coins, with varied success. The public seems quite resistant to the coin version. The mint wants to do the coins since they last much longer than the paper currency.
I was using a vending machine at the local college one night last week. The selection I wanted wasn't in there, so I hit the cancel button and got back one of my two dollars, and a quarter. I was fuming that I'd been ripped off for $.75. The next day I reached into my pocket to pay cash for a small purchase, and the coin I though was a quarter was really one of the new gold colored $1 coins. I had to smile - here I'd been fuming about being ripped off when in fact I'd just been fooled by the new coinage.
Back to the washing dilemma. I use a small bound note book that fits in my back pocket. I swear by these little helpers - I can jot down info about product when shopping, take names/addresses/measurements as needed. They don't need batteries, don't break when you drop them, and are a permanent record. I've even recovered information from them when they go through the wash accidentally... The front loader is usually gentle enough that they come through in pretty good shape. Last time though I used an extra long cycle with hot water, so the booklet was a bit more tattered than usual, and I had to gently separate the pages with 3x5 cards and let them dry out overnight, but I still recovered all the info I'd been gathering.
For those who are curious: Classgear Composition booklet, 80 sheets, 4.5x3.25in, Walmart #W31809.