Currently, 91 countries require US citizens to obtain a visa for entry, either as an attachment to their passport or electronically. Some visas (or "tourist cards") may be issued upon arrival and in some cases payment of an accompanying fee.
US State Dept, IATA, Passportinfo.com and Wikipedia consulted. There is conflicting information for certain countries - some airlines require proof of visa - as described by the IATA.
Links to sources:
Passportinfo.com, https://passportinfo.com/countries-that-require-visas-for-us-citizens/ US State Dept, https://travel.state.gov/content/passports/en/country.html and Wikipedia, https://en.wikipedia.org/wiki/Visa_requirements_for_United_States_citizens
The IATA website with detailed passenger/visa information is currently down.