We/Mailgun focus a lot on incoming mail parsing and routing. Honestly, we find it more interesting than just sending/delivery. Although, there is clearly more market demand for sending. We also allow you to create IMAP/POP3 mailboxes programmatically. We wrote a blog post about it recently[1] and you can also check out our docs for more information[2].

The other companies that I am aware of with incoming email capabilities are Sendgrid with their Parse API[3] and I believe Cloudmailin[4] focuses exclusively on handling incoming email.

Edit: Postmark[5] has incoming email in beta as well. Added links, too

[1] http://blog.mailgun.net/post/12482374892/handle-incoming-ema...

[2] http://documentation.mailgun.net/user_manual.html#receiving-...

[3] http://docs.sendgrid.com/documentation/api/parse-api-2/

[4] http://cloudmailin.com/

[5] http://postmarkapp.com/

What do you want to do with incoming mail? Just outsource the mailboxes to check from your computer/phone, or are you looking for some kind of preprocessing service for connecting mail to your apps?

Check out Context.io, they have a unique and interesting take on "incoming email". Basically, they turn mailboxes into databases.

One I can think of offhand is http://cloudmailin.com/