A friend and I worked on a typical weekend project lately. A Font Awesome to PNG online generator (http://fa2png.io). I was great fun. It took us about 24h total from concept to live deployment. It's a flask app deployed to digital ocean.
We were on HN in 2nd place for about half a day. Got about 17.000 unique visits in the first two days, a couple of twitter mentions and now about 200-300 visits a day.
It was an awesome experience. Very satisfying because it felt kind of complete when we launched. That's not typical for a web app. Although there are always more features to add. We felt like the app is exactly doing what we planned it should.
You could. Not sure if Heroku offers a full fledged linux environment. Also there are unlimited databases (MySQL, Postgres, MongoDB, CouchDB or Redis ) included in the 1€ fee.