You will be able to do SOME THINGS without verification, the question is exactly what...
It's likely just trading that will not require any sort of verification - which is exactly what they stated in very simple terms in their FAQ. I am quite sure that there will be a certain form of verification required in order to have the ability to request a withdrawal of your funds. Everything that BTC-E is saying at this point has really no value. Initially people would be given the opportunity to withdraw 55% of their funds, but nothing was mentioned about losing the latter 45% part in case you do end up withdrawing the offered 55% - this is something they at last came up with, which is ridiculous. Another ridiculous thing is that they convert fiat to the current Bitcoin market rate of $4500, and not the $2546 rate at the time they went down. This just shows that they are doing whatever they can to squeeze out as many funds out of their customers as possible. Every exchange that experienced whatever sort of loss of user funds so far, has at least offered people compensations based on the exchange rate at the day they went down. BTC-E clearly isn't following that 'tradition'.