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Simplified, regardless of your height, the plane would extend to horizontal. Looking around, you would see a perfect half sphere of sky above and half sphere of ground below. Using a theolodite, 180* and 180*.
There are some semantics when you get into when you arrive, how the plane is lit and what the surface material is. If, for example, you and the plan "pop" into existence at the same time, you would see the sky surround you, and then the plane would grow asymptotically from your feet up to the horizon. I don't really care to do the math, but at 1m if you were observing by eye you probably wouldn't notice this growth.
Temperature too, heat can bend light.
But what's more important is atmosphere and just amount of light.
Ahh... Atmosphere. That would blur the horizon to the sky. At very far distances, the scattering and absorption would obscure anything further.