On a tourist visa you get a 30 days time. You could extend that visa only once by visiting the DNATA immigration desk on Sheikh Zayed Road. They will extend it for another 30 days. All you need is to take your passport and ask them for an extension. DNATA visa extension mite cost somewhere around AED 500, as far as I know.
I recently read this article on How to extend Visit Visa
Hope it helps!!