What do you like best?
The mail api is the best tool, you can send mail from isolated hosts whit very realibility, or build mail centric app. the api explain the data, and get traceabiity to all mail acros all recipters
The second "best" if mail gun is the "seller offer" you can build mail complex infraestructure, and make mail secure infrastructre at same time, in single "next based wizard"
And if you integrate whit CMS+mailgun pluggin, you are in the "only" mail server you ned. and exist pluggins to "all" popular CMS.
If you build "by-domain" app, for low trafic, the service is free, and if you build very intensive mail the cost is very rationale.
What do you dislike?
your keys are shared in all deploys, if your need deploy in "high risk" enviremont, and your key are lost, you need change in all deploy points.
The "routes" tool is hard to config, and non intuitive, you can make "no route" rutes and you cant recibe some alerts
Recommendations to others considering the product
nThe mail gun deploy depend of your dns, you need solid dns config to enchance the rewards in mail gun.
You can deploy simple CMS, or complex in house app, is only your desition.
What business problems are you solving with the product? What benefits have you realized?
Send mail from "ondemand" vps for escalable architecture, create a cheap and simple infraestrucutre, whit enchanced security.
You can detect "non-valid email" in advance, and "wait" to open mail,
if is your desire you can build "auditable e-mail" whit great traceability