What size ground rods do I need for a 200 amp residential service?
Ground rod size does not scale with service size. A 200 amp house and a 400 amp house use the same rods. The rule is in 250.52(A)(5) and it is about the rod itself, not the load.
The diameter rule
Section 250.52(A)(5)(b) states that rod type electrodes of stainless steel and copper or zinc coated steel shall be at least 15.87 mm, or 5/8 inch, in diameter unless listed. Section 250.52(A)(5)(a) covers stainless steel, copper, or zinc coated steel rods that are listed, and those may be smaller than 5/8 inch. That listing allowance is why 1/2 inch rods exist and are legal when they carry a listing mark. A plain unlisted 1/2 inch rod is not acceptable.
Practically, use 5/8 inch by 8 foot copper bonded rods. They are the same price at most supply houses, they drive straighter, and no inspector will question them. Half inch rods bend badly in rocky soil.
Length and depth
Section 250.52(A)(5) requires rod electrodes to be not less than 8 feet in length. Section 250.53(G) requires the rod to be driven to a depth of at least 8 feet, and where rock bottom is hit, the rod is permitted to be driven at an oblique angle not to exceed 45 degrees from vertical, or buried in a trench at least 30 inches deep. The upper end must be flush with or below grade unless the aboveground end and the conductor attachment are protected against physical damage under 250.10.
How many rods
Section 250.53(A)(2) requires a single rod, pipe, or plate electrode to be supplemented by an additional electrode. The exception says that if a single rod has a resistance to earth of 25 ohms or less, the supplemental electrode is not required. Nobody tests, so almost everyone drives two rods. Section 250.53(A)(3) requires the two rods to be at least 6 feet apart, and the informational note points out that greater spacing improves performance.
The conductor to the rods
Section 250.66(A) caps the grounding electrode conductor at 6 AWG copper where it connects solely to rod, pipe, or plate electrodes. On a 200 amp dwelling that is 6 AWG copper, and 250.64(B) allows 6 AWG to be run along the surface of the building without a raceway if it is securely fastened and not exposed to physical damage. The connection at the rod needs a listed direct burial fitting per 250.70, and an acorn clamp marked for direct burial is the standard choice.
Field takeaway
Two 5/8 inch by 8 foot rods, 6 feet apart, driven full depth, tied together with a continuous 6 AWG copper conductor and listed direct burial clamps. That passes everywhere.