You could Use localbitcoins and bittylicious to buy bitcoin easily.
For btc-e, I believe you don't need to do ID verification unless you are going to deposit fiat with wire transfer.
https://btc-e.com/page/1Identity verificationRequired in case you choose to load your account by wire transfer.To comply with AML/CTF recommendations, we require our clients to verify identity by providing scanned copy of ID and scanned copy of utility bill or a bank statement which should not be older then 6 month. Copy should be in good resolution and colored.