CloudFlare does not provide a phone number for customer service at this time, save for customers on an Enterprise plan.
The primary reason that we do not do troubleshooting over the phone is that capturing required information is quite difficult through a phone contact, and we cannot verify domain or account ownership verbally, so we cannot provide account-specific answers. Also, sharing useful links and instructions is much easier via email, and less prone to error.
If you would like to get a timely response to an inquiry, please create a ticket and our support team will get back to you very quickly. We respond to every ticket received, in priority order:
Optimized Hosting Partners and Certified Hosting Partners are also CloudFlare customers, and will be helped via email in most cases.