We want to add more than 10 calendars to "Calendar Overlay". This is not supported by SharePoint OOB and we haven't found any alternative (only custom paid components). Is there any component that we could use as starting point? It's on-premises SharePoint 2013, but only client code can be used.
You can use the following methods to display events from multiple calendars in one list.
Content Query Web Part
Content Search Web Part
Linked data source and Data View web part using SharePoint Designer
Using it, you can roll up all items from multiple calendars without 10 overlay limit. However, you need to use code to change the display if you want them to be formatted as Calendar.
Another method to display events from other calendars in a master calendar, you can create an auto-start workflow to create the event in master calendar once a new item is created in other calendars.
Or, use the event receiver to monitor the item events in other calendars and then create/delete items in master calendar.