Jiffies, by Diethard Seiferth, is available in the iOS App Store.
It is a clever and pleasantly effective calendar app which shows your daily schedule both in conventional list form, and also as a traditional analogue clock. This clock does not include hands because it is not telling you the time of day, but rather the time and duration of your events and appointments.
The display is filled in with your colour-coded appointments, on the outer circle for the PM and the inner circle for the AM.
Full-day events are listed outside the clockface.
It really is quite a clever way of presenting the day's committments, showing at a glance what is taking up your time and when you are free for other things, in a way that simply listing a schedule cannot.
You can add and edit your calendar from within the app. You do not need to switch back to the stock iOS calendar for any maintenance. This means Jiffies also includes your calendars for external accounts such as Exchange servers or iCloud.
The app offers a day and night mode, and allows you to change the background image.
Jiffies is a pretty clever app. The only thing I don't really like is that the app uses a badge to show the day of the month (e.g. "6" for the 6th day), rather than the number of events for that day. However, to be fair, Apple does not provide developers a way to modify the app icon in a similar fashion to Apple's own Calendar app which can.