How many alarms can be set on iOS? How many alarms can be set on an iPhone 5? Can you set one for every minute of the day (1440) or is there a limit on the number of alarms?
Well if we want to be 100% accurate, there is a limit. It is the storage capacity of your iPhone 5s. Somewhere that alarm is taking up space and once it runs out, so do the alarms.

I've just tapped the '+' followed by "Save" and got up to over 300 alarms before I realized how much fun I'll have going back and deleting them all. Or,  since their default value after creation is set to "on", absolutely drive people crazy at work tomorrow. 

Also, since it only took 5 minutes, it is a dang good prank to play on a friend - or enemy :)

So, realistically grgarside is right. There is not limit set via programming; however, like I mentioned, the storage capacity is the true cap - eventually at least.