Are there bitcoin exchanges that do not require an extensive verification process to begin trading? I do not want to submit forms on MtGox.
No U.S. exchange. Btc-e does not but they will not do business with U.S. banks. Coinbase does not have an extensive verification process, but they are technically not an exchange, and to buy or sell bitcoins through their service you have to link a U.S. bank account which for all intents and purposes is the same thing as verification.
Due to the patriot act any currency exchange doing business with the U.S. has the burden of proving that they are not laundering money for terrorists.
Most exchanges need your identity information. If you do not want to submit that information, your best bet is face to face or OTC trading with other bitcoiners. If you build up a good reputation trading OTC, then you can trade freely with trusted contacts, but the amounts you can trade will probably be smaller, and at a premium.
Online fiat exchanges can be heavily regulated. To avoid this inconvenience, when you are for example just buying small amount, you can buy over the counter from a friend. Simply send or give the fiat to a friend and recieve the bitcoin.
Another method is to use a bitcoin cash machine.