I've started doing some chalice dungeons in Bloodborne and I'm a little uncertain as to what to do once I have cleared them. There doesn't seem to be much help on it in the game (like everything else) so I'm not sure what I should be doing when I've got to the third level and killed the boss (And explored all side areas).
I have the option to remove the chalice but it occurs to me there might be a reason why it doesn't just go once you have completed it. Is there a way to restart it? Is there a way to do anything else? Or once I've killed and looted everything do i just remove it and start a new one?
You can save dungeons to have quick access to Bath Messengers's shop which can be used to purchase weapons. Not every dungeon has this shop however.
There are some relatively good gems that drop from respawnable enemies. You can keep the dungeon to farm those whenever you want. (I've covered farming in some of my previous answers - here and here). This is very helpful if you don't have PS+ subscription and cannot just open one of the best possible dungeons (i.e. with bosses that drop the best gems) via glyphs.
You can farm blood echoes there if you got good mob placing right next to your lamp.
You can also farm materials required for dungeon creation (Ritual Blood and all the other stuff) in dungeons of any depth. Thanks @Ben for this one.