As far as I'm aware, Oyster's aren't ted to individuals until you sign up for a season ticket on them, which seems to be anything longer than a week travel card. Just buying a card and using cash pre-pay top ups should be anonymous (although the card itself can still be tracked). I'm not sure how easy it is to tie a card to a person if you buy it by switch or credit card or similar without signing up for a season ticket though.
Been thinking about some sort of Oyster exchange pool where people top up a card to a pre-set amount then you all put them in a pool once a week and take a lucky dip for a new card. Or could just pony up the extra £3 a week for a new card I guess.