Electrical – What size wire do I need for a 600ft run

First, figure out if it's even your problem. As commented by @jwh20 , the "service drop" is often (but not always) the power company's responsibility. Whether or not that's the case, it has to meet the power company's specifications. i.e. for that distance, the company might install two additional poles and a transformer, you write them a large check for installing that, or a smaller ongoing payment on your power bill for some years, the job is done.

Speaking from personal experience, my power company would only install a 100 foot or less overhead service drop, but was willing to use "customer-owned wire" in an underground configuration from the pole to their meter at the house (inspected and approved by them.) We ran conduit to the base of the pole, leaving a coil of far more wire than was actually needed (but exactly what they specified) and conduit for them to go up the pole with. Other utilities may operate differently, such as putting the meter and a disconnect at the pole, which changes things.

While I dug the trench myself and backfilled it myself, I had a licensed master electrician on the job to make sure that the thing was done correctly (and it kept the power company happy.) It's not exactly a great fit for fully DIY installation.

On the low end (assuming the usual 75C connections for large ampacity gear), you would need 4/0 aluminum (you get to treat 200A feeding a house as 166A for ampacity, per NEC) but at that distance you'll likely want more like 250-350 mcm aluminum for voltage drop. Whether you need 3 or 4 wires depends on where the service disconnect (and ground-neutral bond) is located - if it's at the pole, you need 4 wires (separate neutral and ground) - if at the house, only 3.

Since you have clarified that you are also looking at underground service, I will recommend a few things beyond wire. Use Schedule 80 PVC conduit - you need it at the ends for "protection from physical damage" anyway, and it does not cost much more than schedule 40, but is much tougher. Don't use direct burial cable - it's cheap the first time, expensive when you have to dig again to replace it since the cable is exposed directly to any insult (sharp rocks, rodent teeth, etc.) Do shop at a real electrical supply (you won't find schedule 80 at HD/Lowes) and if you can get 20 foot sections from them, do. Deburr the male ends of all conduit sections. Be sure to grab a roll of "buried electrical line below" tape. Place an empty "communications" conduit in the trench - trenches are expensive, conduit is cheap, you don't want to have to dig again when the fiber to the home provider shows up 12 years from now (or whatever.)