Since money and availability are listed as criteria (I didn't make the rules) that would narrow it down to whitetails or coyotes. Probably coyotes because there's no closed season, no limit, you can call and trap them too, and much of the time you might be able to make gas money. They are also perfect for a gun nut, and precision/long range shooter. Since its a year round thing weather can be whatever you want it to be. Hunting techniques tend to be active, so you don't have to sit for days waiting for something to happen.
Take money out of it completely and it would be elephant. Exotic, active hunting style and they occasionally try to kill you. Its impossible to be bored when you're close to an elephant. Great sport for people who like big guns. Nice weather if you like hot.
If money is a little tighter but not a major problem then buffalo. If you lump bison in there, then you can hunt them on every continent. Hunting techniques tend to be alive and mobile, they are good big gun opportunity for a gun nut and they occasionally try to kill you. Weather is predictable.