iPhone 5 alarm—is there an option to have an “increasing melody” for the alarm?

Solution 1:

The alarm will not activate when the iPhone is turned off. This was tested with an iPhone4 running iOS6.

Solution 2:

As Pro Backup states, if the iPhone is off, it will not turn on by itself to activate your alarm.

And as far as the stock alarm built into the iPhone, there's no option for you to enable a gradually increasing volume alarm. However, a quick Google search found the goodNite Lite - Alarm Clock Night Light app that should give you the functionality you're looking for in that department.

Solution 3:

I've been using Yocto Clock for my alarm. It has an increasing volume that can be adjusted up to 30 seconds. As far as turning the phone on goes, I don't think it does that. It does, however, use the proximity sensor to enable a kind of sleep mode. This allows you to turn the phone over, screen faced down, so it sleeps. The alarm still goes off, and picking the phone up enables the other features (like built-in flashlight).