l am a British citizen with a British passport and, in my passport, I have many, many visas for Syria, Lebanon, Egypt, Zanzibar and other countries in the Middle East. I am flying with Emirates from Athens to Newark. Will the United States let me enter?
The US does not automatically deny entry based on which countries you have previously travelled to.
British citizens are generally eligible for the Visa Waiver Program when visiting as tourists. However, you are probably not eligible. Quoting from the US Department of State concerning exceptions to Visa Waiver:
Nationals of VWP countries who have traveled to or been present in Iran, Iraq, Libya, Somalia, Sudan, Syria, or Yemen on or after March 1, 2011 (with limited exceptions for travel for diplomatic or military purposes in the service of a VWP country).
So if the above applies to you, you will need to apply for a US visa. Given your travel history, you should expect to receive a higher than usual degree of scrutiny from US authorities.
Note that all travellers to the US without a visa, hoping to enter the US under the VWP, are required to have instead have an approved ESTA; otherwise you may be denied boarding to the plane. If you think your travel has not disqualified you from using the VWP you should instead apply for an ESTA. If you are approved you should be okay to travel that way.
You will most likely need a visa.
However, my advice is you try getting an ESTA, the mandatory passport registration required for most visa-free nationals (including British) entering by air or sea. Answer the question regarding you having visited Syria truthfully, and add any explanations necessary if offered to do so.
In the unlikely (but not impossible) event that you're approved, you're good to go, otherwise you need to apply for a visa.
Do note that neither an approved ESTA or visa is a guarantee of admission.