How many years can the IRS go back to demand payment of an unpaid balance of assessment?

Question Details:

I have not filed a tax return in over 14 years. I recently a letter form them(IRS) claiming I owe nearly $60,000 in unpaid taxes for the years 2004, 2006, 2007 and 2008. I am planning to negotiate a settlement with them to try and clear the matter. Once I have settled with them for the 4 years they say I owe, will that be the end of it or will they pursue me for the other 10 years I didn't file?

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