Really? Nobody knows how to give a user the ability to add a new module (HTML or whatever...)

Module Deploy permissions are granted to non-administrators on a per extension basis.

Navigate to Admin --> Extensions and you will see a list of installed modules. Click on the edit pencil beside an installed module and you will see the Deploy Permissions grid. Select the role(s) to which you will grant permission to add a new module instance on pages where the role also has been given page edit permissions.

I'm surprised that uses in the non-administrators role were able to add existing modules without having been granted Deploy permission.

EDIT: Sebastian - You type faster than I do :)

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Bill, WESNet Designs
Team Lead - DotNetNuke Gallery Module Project (Not Actively Being Developed)
Extensions Forge Projects . . .
Current: UserExport, ContentDeJour, ePrayer, DNN NewsTicker, By Invitation
Coming Soon: FRBO-For Rent By Owner