How low can you go....legally?

I'm no construction engineer but I would imagine that were I to want to build myself a cellar or basement under my house, provided I was sensible and careful about gas pipes and water mains and such like, I would be entitled to do so. But how far down is the maximum I could dig? Would I need planning permission or can I construct a 70 floor basement on a whim with no permission? And what would it be called? A skyscraper goes skywards so what would this be called? I propose naming it a 'Corescraper'....or as UnterTower...or a TectonicTower or CrustBuster...

Hmmmm. There is such a thing as air rights for the space over land that you own. I don't know if they are called "ground rights for the area underneath. The laws that you would want to look at would be under mineral rights. That's a long established area of law about digging/excavating on land that you own.

If you live in an unincorporated area, I think you can just dig down to "the center of the Earth".

If you live in a city I'm sure there are plenty of city codes, and requirements to get permits.

im very interested in this idea also,im in love with the idea,anyway you could iether keep it secret(not likely with such a size) or get council approval,which is not that difficult,you just have to declare the dimentions,and prove that it wont be a issue for neighbours,or any tennants you may have residing there,and have an engineers report done at intervals,engineers reports can range from $500 to thousands,maybe start a bit smaller,your idea is fantastic!

It depends on what country and state you live in. Do you live over a mine or oilfield for instance?. There may well be laws governing your project, I suggest you start by checking out with your local authorities.
By the way how do you propose to dispose of the spoil from your digging?.

Talk to an engineer...He would know all the answers and I would not suggest taking a project on like this for a do-it-yourselfer! You have to have the proper permits!