For vias, are the annular rings only for the layers which you would like to connect? If this is true, can I remove the annular ring for layers the via doesn’t connect to gain extra clearance or move traces just a bit closer to the via?
For example, routing a signal from the top layer to layer 3, would layer 2 and the bottom layer need a ring as well? I have no idea what happens in the manufacturing of boards for this. 
I’m trying to route a bga for the first time, and I’ve never thought about what annular rings on vias are actually for. If I leave them, I don’t have room to route. It seems like Altium allows you to choose the annular ring size per layer. 
I’m trying to route a 0.8mm xilinx fpga bga with 4mil traces and 0.2mm vias. I need a 6 mil annular ring by manufacture’s spec for lower cost boards. If I use the advanced production the boards cost twice as much.  
Thanks for the help guys!

Here’s the settings for a via in Altium.