Are there any triple-changing Transformers that possess both a beast mode and a vehicle mode?
Sure! There’s the G1 Horrocons, Apeface and Snapdragon, who each turned into a jet and a creature (an ape and a dragon, respectively, surprising nobody):
You might also count Scorponok, who had a scorpion mode and a city mode, but then a city isn’t a vehicle, so!
There’s BWII Galvatron, who turns into a dragon and a drill tank:
Just recently, there’s Titans Return Alpha Trion, a spacecraft and a lion!
Or how about the unproduced G1 Triple-Changer that turns into a jet and a horse, who was later brought to life in the Armada comic as “Dropshot”:
Going beyond three modes, there’s always G1 Sixshot, who has a wolf mode buried amongst his many vehicles, or Quickswitch, who had a puma mode. RiD Megatron has ten modes, half of which are beasts: a dragon, a gargoyle, a mammoth, a pteranodon, and a griffin-thingie.
I’m sure there are more!