Q: Which is the biggest star out there, if biggest means the greatest radius?
A: The stars with the biggest radii are generally considered to be red supergiant stars. Some folks define a class they call "hypergiants," but most astronomers do not split hairs that finely. On the Hertzsprung-Russell diagram, lines of constant radius run diagonally from upper left to lower right. Hence, stars in the extreme upper right will be the ones with the largest radii. Such a star will have a low temperature (red color) and a high luminosity. To achieve such high luminosity while having a low temperature, it must have a huge surface area — thus, it will be a red supergiant.
Well known red supergiants include Betelgeuse and Antares, but neither is the largest. I don't do stellar research, so I'd have to look up what others say. An article on Universe Today says the largest is VY Canis Majoris (VY CMa), measuring about 1800 times the size of the Sun. A paper by Emily Levesque (University of Hawaii) cautions that VY CMa has a wide range of size estimates, but agrees that the largest stellar radii are about 1500 times the radius of our Sun. The paper lists four such behemoths: KW Sagittarii, Case 75, KY Cygni, and Mu Cephei. If placed in our solar system, these red supergiants would fill space almost to the orbit of Saturn. The entire orbits of Mercury, Venus, Earth, Mars, the Asteroid Belt, and Jupiter are all smaller than the diameters of the largest stars.