Generally you can’t serve cookie-free content while using its CDN (Reverse Proxy) services together. The way Cloudflare provide services in any plan, it must adds a special cookie namely _cfduid with each HTTP requests. For your kind information, there is no official way to eliminate these cookies regardless of any Plan unless if you stop using its CDN services. You may continue using fastest DNS services without any issue.
Why excluding cookies with Cloudflare is so tough?
Here’s the answer. I have copied this screenshot from my browser network timeline. Look at path. Cloudflare is forcing cookies for domain as well as subdomain. This is why it’s tough to skip it.
If you love using Cloudflare, here’s the workaround to bypass _cfduid cookie of Cloudflare for the static resources .
Step 1. If you really love Cloudflare, Keep using its DNS & CDN services for main domain.
But to exclude cookie for the static resources you have to change the hostname completely.
Step 2. Sign up for a CDN services, grab its naked URL (or masked it with new domain) since our purpose is to change hostname; and enter into CDN path into your Cache plugin.
Check at GTmetrix, Voila! There is no more any cookie for the static resources.
There are few more steps, you can learn at below article.
Note: I won’t recommend tweaking so much. Unless you are feeling crazy for getting unrealistic 100/100 score.