What is the maximum amount of beer I can transfer into the Brew Bucket and ferment?

The total capacity of the vessel is just a hair shy of 7 gals.  

The answer to this one really depends on whether you are making an aggressive/high gravity big beer or a more mild-mannered beer.

This is what really dictates how much head room your beer will need. And if you are planning to dump in 5.5 or 6 gals of beer AND you plan on having a big aggressive beer in there, then you would probably want to run a 1/2" ID silicone tube as a blow-off so that you don't get a big mess through your bubbler in the event of a beer volcano type of event!  

Always best to use a blow off tube in the first few days of primary and our lid hole on the Bucket is perfectly and intentionally designed to take a 1/2" silicone tube stuffed right into the hole.  

We sell the tubing on our site in the accessories ( 1/2" tubing  ) 

We also sell a barbed blow off elbow for use with the brew bucket lids. ( Blow Off Elbow )