Depending on the country, you’d have to go to some local office that handles alien affairs, and apply for an extension of your time there. But spending your time in bureaucratic offices doing paperwork is probably not how you would like to spend your extended European holiday. You can also play dumb, and overstay your 90-day visa-free stay, but you risk getting slapped with a fine and a stamp that won’t allow you to re-enter for a very long time (3 years, I believe?). Both of these are rare, and some coutnries and/or customs agents enforce it more than others.

My advice is, either plan not to stay longer than 90 days, or [if you seriously anticipate that you’ll be there more than 90 days] take care of the paperwork beforehand and apply for a Schengen Visa.