Photography Equipment for Shooting Brown Bears in Alaska

What will I need?
Obviously an excellent question but not one with an obvious answer.  Short version would be the best you have, but here is a more informative list.
The two items you really can't do without is a camera that shoots well in low light and at least one long lens.  
It's going to be cloudy most of the time, and you will be shooting over bright water so you will have to overexpose, usually more than one stop.  Plus to bring out the bears sunken eyes it's usually best to push at least another 1/3 stop.  Hence to get the speed I need for a running bear in the water and to stop the action I'm usually set at ISO 1600.  If you have a camera body that can handle that speed without grain, you are good to go.  If not, I'd really suggest renting one for a week.  You're not going to get this opportunity often and you want to make the best of it.  Many bear photographers I know shoot a higher ISO than 1600 (3200 for a few) so I can't emphasize that enough.
What size long lens is harder to answer.  However, for the Katmai Trip I think a 500 is the ticket.  I've had people come with a 400 but I think they were a little disappointed. Especially since most cameras that shoot high ISO are full frame you will need distance "most of the time".  I personally like to have another camera close by with a shorter lens, I use a 70 - 200 with a 2x.  But the new Canon or Nikon 1 - 400 are much sharper than the old ones so they are great as well.  Teleconverters are great, as long as your camera body can handle the loss of light.  We will get close, too close for a long lens at times but I try to keep those encounters to a minimum. 
I had one person use a walking stick instead of a sturdy Tripod, but I would strongly suggest a sturdy tripod that is somewhat heavy.  You will be setting it up in moving water and you want it heavy enough that the current does not take it away.  Most graphite tripods are fine as long as you have a heavy Gimbal type camera mount that will let you swing with the bear.  I have a ball head that claims it's a gimbal mount but I would never use it for my camera in the water, not sturdy enough.
Filters such as a polarizing lens or ND are not really needed.  It would be nice to cut some of the glare from the water but I don't think you want to lose that much light.  I like my shutter speeds to be at least 1/400, and 1/250 at the lowest.  Rushing bears often blur at slower speeds.
Even though the larger camera companies claim much of their gear is water or weather proof I think it's silly to tempt fate.  I arrive with at least 4 rain sleeves to cover my camera and lens.  You can spend from $8 to $180.  I personally go with the $8 ones, and have not had a problem.  See this item on BandHphoto.com http://www.bhphotovideo.com/c/product/1022872-REG/op_tech_usa_9001252_25_mega_rain_sleeve.html
They work fine for me but if you want a better one go for it.  I don't have a lot of need for rain sleeves in AZ.
You also will need a good sturdy camera bag, preferably one you that has a backpack type feature.  And it needs at least a rain poncho over it to keep it and your gear dry.  Each day we will leave the lodge and drive to the float plane dock, there you will load your gear in the plane.  When we land we may have as short of walk as 25 yards, or one as long as half a mile.
I think a helpful list for you to decide what to take would be to look at my own packing list
Camera bodies
Canon 5D Mark III (high ISO but full frame)
Canon 7D Mark II (not as good in low light but a superior video DLSR and I'm moving in that direction more and more)
Lenses
Canon 500 F 4, (I've had clients bring an 800 and were happy with it but it's not necessary; If I was going to rent a large lens it would be a 600 only because my best high ISO camera is full frame)
Canon 70 - 200 2.8 (I will almost always have one of the below on)
1.4 and 2X teleconverters
24 - 105 F 4.0 (I do like to get some landscape shots of the areas we visit as well).
I carry all of this in a LowePro backpack bag which is my "older version".  It's tougher, waterproof with a rain cover, and I don't care if it gets muddy, because it will.  My more often used Think Tank Traveller stays at home to stay cleaner.
I hope this list helps anyone that reads it, but most importantly any one that comes with me.  If you are on one of my trips, and you have any questions please drop me an email with your phone number and a time I can call so I make sure you understand what I'm suggesting.
Good shooting.