Ben- I think it depends a tonne on what you are looking to do. If you want to settle in one city (maybe a sanctuary city), make money, and live under the radar, you can get away with it for a long time. If you think to travel, consider the occasional dip across the border to reset you visa (or, I'm assuming you have this ESTA thing coming from UK?)
I had a friend who was deported in 2008 I think, and it was fucked. We were partying for new years in Slab city, and when leaving, the Bus she was on was controlled (I guess 100+Km from the border). She was held in ICE detention (fucked black prison) for just over 2 months, and only let out so soon because friends payed for her trip home. The flight back to France (her country of origin) was 1800 euros, and she will not be allowed to return for 10 years.
So, it depends a lot on where you are, I don't know what happened with Arizona's immigration laws, but just because they are designed for Mexicans, doesn't mean they wont bust you too.
Crossing the border into Mexico (last time I did was 2008, so correct me if Im wrong), is a cakewalk, no controls, just a turn style gate to walk through. walk in a circle and return to the US side, tell em you have been in Mexico for the last months traveling. I think the ESTA is good for 3 months of every 6, for 2 years, and only costs 12 euro or something. I guess this should be the same for you, I was just looking into it for my partner, she is from Germany though.
Coming from the UK; you have some of the best travel privileges in the world (actually, THE best I believe), meaning you can enter almost any country without a visa. I guess this mean Mexico, so lacking a stamp in your passport shouldn't be an issue upon returning to the US.
They are fucking serious though, be careful. If my french friend hadnt gotten someone to pay for her flight ("voluntary deportation"), the crime of illegally immigrating is a felony, and you can expect a year in black prison before they get around to flying you home.
In my opinion, now is a bad time to be fucking around too much, as it seems things are on a downward spiral for people rights in the states. I have been abroad for over 2 years now, and I cant say from experience, but from what I hear, I'm scared to return, and I am a citizen.