Chimney additional height

Came go a additional 4 courses up in the chimney.

Technically, no reason you couldn’t, though I would be a little concerned that the chimney is not really tied in to the structure other than through the mortar base. The three courses are nicely balanced on the structure, so they are not a concern. But seven courses of brick gets tall and tippy, not to mention more weight above the base course.

If your goal is to get some more height, another option is an extension on your 6-inch pipe. The advantages are lighter weight and lower profile presented to the wind. Another possible help is that bringing your chimney cap higher might improve draw from your oven. I’m cautious about that because there are mixed experiences with that from other users.

Hope this helps, Rob!