You can purchase bitcoins from traders in any P2P marketplace, as these exchanges do not require verification of any sorts. In LocalBitcoins, for instance, there is an option “ID not required” where you would list traders that would sell you bitcoins without asking you to verify yourself.
Buying BTC anonymously
You can also use ATM machines near you to purchase bitcoins without providing personal details.
Bitcoin ATM machine
Lastly, you can use multiple exchanges that allow very small purchases without verification.