Up until recently, AMD's burliest consumer CPU was the Ryzen 9 3950X. It has since been replaced by the Ryzen 9 5950X at the top of the stack, though if you want to save a few bucks (and build a PC right now, rather than wait for Zen 3 inventories to normalize), the Ryzen 9 3950X is a compelling option at its current discount.
You can get it for $694.99 at Newegg currently, after applying coupon code 93XQC55 at checkout. As an added bonus, it qualifies for AMD's "Equipped to Win" game promotion, which gifts you a code to download Far Cry 6 when it releases next year. For whatever odd reason, the game promo is only valid with Ryzen 3000 series purchases, and not the Ryzen 5000 series.
This is not a gigantic savings, but if you need and/or want 16 cores and 32 threads of computing muscle, it does not get any cheaper than the Ryzen 9 3950X. And at this discount, it is more than $100 less expensive than the 16-core/32-thread Ryzen 9 5950X. Whether you should wait and splurge an extra Benjamin on the Zen 3 part is up to you, but you could certainly make an argument for the Ryzen 9 3950X if wanting to build or upgrade today.
I would not necessarily recommend either one strictly for gaming, though they are suitable for that. Instead, these are some of the best CPUs for gaming and serious streaming combined, or any tasks that will tap into all of those cores and threads at their disposal (like video encoding and other content creation workloads).
Incidentally, the Ryzen 9 3950X is even cheaper at Ant Online, where it can be bought for $679.99. It's not clear if that listing will qualify for the Far Cry 6 freebie, though. In theory it should—AMD lists Ant Online as a participating retailer—but there is a separate product page, that specifically mentions Far Cry 6, for $759.99.
Update: Ant Online clarified to us via email that "any customer who purchases the 3950X from us will be receiving a code for Far Cry 6." Additionally, the retailer updated the latter product page with a specific mention to the offer, to reflect the $679.99 price (instead of $759.99).
