I had to make a lots of tests in order to pin out this issue on my side.
The "no password" feature did work well until I started to mess with restrictions....
I am currently setting up an IPod 5 for my child, and this issue show up when I activate restrictions AND restrict the "Apps" config. Restoring this config to it's original value did not fix the bug. So this is a one way to the issue.
On my side, when I got this problem, I had to reset settings AND content to start up with a brand "new" IPod. My entire apple identity is new so I didn't have anything to loose anyway. It did not work the one time I tried to only reset the settings.
So here would be the trick to avoid the issue:
First setup your iOS to allow a "no password" for free apps. This should work.
Then activate the restrictions.
The following restrictions configuration were setup without triggering the issue:
(Sorry if the terms are not exactly the same, my display is in french actually...)
0 => Did not touch that configuration
1 => The feature was restricted
Authorize
----------------
Safari - 0
Camera - 0
Siri - 0
Facetime - 0
AirDrop - 1
iTune Store - 0
Apple Music Connect - 0
iBooks Store - 0
Podcasts - 0
Add apps - 0
Remove apps - 0
In-app purchases - 1 (Of course, who wouldn't ?)
Authorized Content
---------------------------------
Classification - 1 (Canada, on my case)
Music, podcasts,etc.. - 1
Movies - 1
TV - 1
Books - 1
Apps - 0 (THIS ONE TRIGGER THE BUG IF MODIFIED)
Siri - 1
Web sites - 1
Password Config - 0
Privacy
---------------
Localisation - 0
Contacts - 1
Calendars - 1
Reminders - 1
Photos - 0
share position - 1
Blutooth - 1
Microphone - 0
Twitter - 1
Facebook - 1
Ads - 0
Authorize modifications
----------------------------------
Accounts - 1
Background updates - 0
Max volume - 0
Game Center
-------------------------
Multiplayer games - 0
Add friends - 0