I've used it in the past for an office phone system with DID delivered over IAX, Linksys SIP desktop phones, and no usage of the API.
Also separately for an inbound call centre with simple routing based on customer type and language(s) spoken by team members. It worked well, and I remember paying $50 or $100 or similar for a license for an add-on which displayed the live status of each queue in a web page, which we put up on a large screen. It worked well for many years with very minimal maintenance. I experimented with some basic outbound auto-dialing features: take a CSV of phone number and customer ID pairs. Dial each number until one connects, then transfer the call to a member of the team, and pop up the customer ID on their screen, so that they can paste it into our CRM app.
Today I only use it for my personal conference call server. Not using MeetMe, but whatever the newer conference bridge is called.
I like asterisk, and glad to know it now has a REST API.