How to make part of rigged mesh stationary? How can I make a few specific vertices stay in place and not be affected by the rigging? I've messed around with weight painting and no matter how blue I make it for all the bones it still moves. It actually moves even more when I make it bluer which doesn't make sense to me. Thanks for your time. 
I've uploaded my file so you can see what I'm working with. I need the base of the tentacle to stay put. Just the first layer of vertices that are moving a little bit.

I'm using Blender 2.8
As you can read in edit mode, your base vertices do have some weight relatively to many bones, so of course they are moving.

If you want them to stay still, select all of them and remove weight from all vertex groups.

Another more usual way is to create a "tentacle base" bone and weight them to it.

Remember that even a very little weight, if it's not "counterbalanced" by a bigger weight, will have a great influence to the vertex, so you can't trust the blue color in weight paint mode only.