The grandaddy of Tequila styles is called Anejo Tequila. Anejo Tequila is aged a minimum of one year and sometimes for several more years depending on what flavor profile the manufacturer is going for. This is opposed to blanco and reposado Tequilas which are either aged only a number of months or not at all.
Anejo Tequila lends itself well to sipping by itself. There is a lot going on in anejo so sipping tends to be the preferred method of enjoying it. Some cocktails (besides the ever present Maragarita) take to anejo Tequila rather nicely as well. An example is an Old Fashioned which you can make with anejo using a lime peel and a say grapefruit bitters instead of aromatic bitters.
Unfortunately anejo tends to be the most expensive Tequila as the extra aging costs money. But it's usually not that much more than blanco and reposado. If you're already in $35 territory with the blanco, another $8 isn't going to kill you if you are looking for the anejo.