I know this picture is from what was a mediocre video game [Lionheart Kings Crusade] the likes of which doesn't really aspire toward historical authenticity. But as a newfangled great helm-wearer, I can't help but go back to this picture a thousand times.

It could just be me. But I doubt he'd be wearing Wisby-style gauntlets.

I have been reading Arms & Armor of the Medieval Knight by David Edge and John Miles Paddock and I know that in the 13th century, the name of the game was mostly maile. You started seeing things like the great helm, and plate armour with things like poleyns, cops, and coats of plate. I daresay most pictures I've seen and almost all descriptions I've read infer heavily that the mainstay neck protection whilst wearing a great helm was an aventail attached to a bascinet underneath.

...but is it possible a coats of plate-style gorget might have been worn with it?

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I've seen no evidence for it - neither written nor pictorial (c13 anyway - I don't really look at c14).

c13 neck/shoulder defenses generally consisted of a collar of some kind usually under the mail coif. (but sometimes seen over or even alone on poorer infantry).

There are accounts of Baleen (whalebone) collars being ordered in for tournaments, but not sure if it was the same for the battlefield or if it was standard gamboised armour.

Also for that period, you see the advent of the cuiries, armoured surcotes, coats of plates, all of which provide extra coverage to the chest/collarbone area.

So we have the greathem covering the front of the neck (they are shaped specifically for this), the chest/collarbone area has (from bottom to top) aketon, mail hauberk, coat of plates, mail coif. I think that's plenty. And neck protection was really a case of filling in the smallish gap between that and the greathelm - mainly for the sides/back of the neck. I'd suspect that a strong textile or composite textile/whalebone collar under mail is enough for that task - since most blows landing there are already likely to have glanced off the shoulder or the edge of the helm anyway.

It would feel like overkill to me to have it as an extra coat-of-plates style defence under the mail coif as well.