For plinking it's hard to beat a .223. There's plenty of ammo types available and it's far cheaper than most other options. If you're not reloading, plinking with a rifle that costs $2-$3 per shot can get pretty expensive!
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My plinker setup is a Ruger Predator in .223. I've run about 1400 rounds of reloads through it in the past year, and I've been amazed at how accurate it can be at its pricepoint. With 68 and 77 grain reloads it's MOA all day, even when running warm. It's plenty for reaching out to 500 yards, and can teach a lot about ballistics and wind holds. I mounted the action into a Magpul Hunter stock and added an extended bolt handle and trigger spring kit.
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I had the Predator with me today, but by the time I moved over to the long range a bank of fog was rolling in. It's tough to shoot at 500 yards when you can't see past 200!
If you go with a cheaper option like this one you'll probably end up adding a rail, changing the stock, and maybe modifying the bolt handle. All that adds up! One of the other rifles I usually bring to the long range is a 6.5 Bergara HMR. At under $1k, these come with a great stock you won't want to replace, a 20 MOA rail, and extended bolt handle. The bolt is just as smooth as my Tikka T3X, and has the lightest lift of any of my bolt guns. Plus it shoots amazingly well.