A surprising number of companies do include “you may not use the service we provide you to compete with us” in their terms of service.
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After a quick search the best example is Atlassian. It would (apparently, IANAL) break terms to plan a JIRA competitor using JIRA.
> Customer must not (and must not permit anyone else to): [...] (d) use the Products to develop a similar or competing product or service
https://www.atlassian.com/legal/atlassian-customer-agreementAlso Salesforce. Their competitors are explicitly disallowed from using any of their services for any reason.
> SFDC’s direct competitors are prohibited from accessing the Services, except with SFDC’s prior written consent.
https://www.salesforce.com/en-us/wp-content/uploads/sites/4/...