The one I would recommend is usually about twice your budget in good condition, but it is a very nice pro grade tele zoom. I'm talking about the AF-S NIKKOR 80-400mm f/4.5-5.6 G ED VR.
If the budget won't stretch that far, the AF-S 70-300 VR worked really nice on the 10, 12 and 16 MP DX sensors, but gets a bit mushy from 230 mm or so on the 24 and 20 MP DX sensors. The corners look nice on 12 MP FX, but it shows its age already on the D4 sensor and it gets more visible on the later, higher resolution sensors.
A first-gen 70-200/2.8 can be found for around $700 or so in good condition and takes the 1.7X converter well if you need more reach after using it for a while. It predates the FX bodies by some years and vignettes a bit much on FX.
Keep away from the screw-drive 80-400 as it focuses slow as molasses and also keep away from the AF-S 80-200 as there are no spares available to repair them when they wear out.
The third generation of the 70-200 has electronic aperture actuation and won't work on a D80. The AF-P lenses aren't compatible either.
The photos in this post from 2016 are all shot with an AF-S 80-400 on a D300.