As others have pointed out, most casino cages will sell you a $1 chip. But if you can't get one that way, you can always buy in at the blackjack or craps table, look through the stack for the best one, then cash in the balance at the cage. Some casinos use $1 tokens instead of chips on the BJ table; if so, check the craps table for $1 chips. Just place a $20 on the layout and ask the dealer for "change only - stack of $1's please". You can inspect the chips at the rail of the craps table, then leave after pocketing the ones you want to keep.

It's quite an ordeal to make it to ALL the casinos on one trip, especially all the outlying ones. For those, you might consider purchasing them from a chip dealer such as Cheques in the Mail or from Andy Hughes' Nevada Casino Chips. Regular table chips are usually $2 each from such chip dealers.