Suppose you took earth and gave it approximately 3 times the air, with most of the extra being N2 or Ar.
Combustion goes up with the partial pressure of oxygen. So unless we want to burn down the planet around our ears, at 45 psi, we want only 7% O2 or so. This much Nitrogen starts implying nitrogen narcosis, but we will assume that the natives evolved with this pressure, or that visitors can use medication.
Three times the atmosphere will still be fairly transparent. There will be some additional heating in the air from absorption of sunlight, but the bulk will be at the surface.
Tropopause will be much higher.
Three times the air will mean 3 times the water vapour. Days that are now thin cloud would be heavily overcast. Heavy overcast would become dim twilight.
This may reduce sunlight that reaches the surface, resulting in heating that takes place more evenly through the atmosphere.
Stronger greenhouse effect? Move the planet out a few million km?
Rainfall intensity could be brutal.
A rising parcel of air would have much more energy available, and the density differential would be greater. This would mean either a smaller temperature differential would be needed to separate a parcel of air from the surface, and/or updrafts would be incredibly violent.
With more turbulent weather, lightning would be more frequent. Updrafts could support both larger raindrops and larger hail. Being outside in a hail storm could be lethal.
Wind would have 3 times the force at a given velocity. Dust would be picked up at lower velocity, and due to increased drag, would take longer to settle out of the air.
Any other thoughts on the effect of air density on weather?