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Sendgrid is a great example of an API-only startup. They've sent 45 Billion transactional emails for over 60,000 customers.

Along similar lines:

http://www.mailgun.com/

http://postmarkapp.com/

And a couple of non-mail-related ones:

http://www.fliptop.com/ - social intelligence API

http://www.twilio.com - SMS & IP Telephony API

The Echo Nest (http://the.echonest.com/)

Mashery would be another example.

Often companies that have a strong business case for developers using their APIs will sponsor hackathons, so you could find more such companies by looking at lists of hackathon sponsors.

http://phaxio.com

http://www.feedsapi.com fits well in this category

NewsCred (http://www.newscred.com)

https://www.opencnam.com/

I think Kinvey fits into this category. Backend as a Service.

Urban Airship. Parse.

Twilio?

Tropo

Zapier?

Love Zapier, but I'd say they're basically the opposite... They have a product based almost exclusively on external APIs.

Pusher?

apiary.io

janrain?

Ask HN: API-only startups?