I'm looking for a cron job web service that is:
I need to be able to schedule cron to execute.
Azure Scheduler can do HTTP requests (GET/POST) and has a free tier. It is very reliable and can be bent in any way. Free tier is limited with number of requests, but can be worked with.
Also there is Aditi Scheduler that has a free tier limited to 500 requests/month. I have not tried it, so can't comment on how flexible it is.
This might be an example of where you can't have all 3 characteristics; you can only pick 2.
Try https://uptimerobot.com/. It's not that flexible (can't schedule specific times), but you get the added benefit of being notified when something fails. If that doesn't work, you can try user2813274's suggestion of Google App Engine.
EasyCron https://www.easycron.com provides the most stable Cron Job Services. Here you can schedule cron jobs with execution logs, Email notifications, run time predictions and a bunch of other featuers!