Recently did a mock auction on ESPN.com (12 team mixed redraft), and the following players went for $1 or $2. Every league is different, so I can't promise that these players will be available in your league for $1-$2. But if they are, these players are great targets.
I list them here by position, in order of descending value. For $1-$2 players in a mixed league, I tend to value upside over experience: