I need to solve convective dominated flows problem. If high order polynomial (chebyshev/Legendre) will be used as basis function, which is the best, Discontinuous Galekin (DG) or Streamline Upwind Petrov Galerkin (SUPG)?

Use Finite volume: SUPG is unbounded and DG unbelievably expensive (and from what I hear, the state-of-the-art is such that is still blows up in your face from time to time for things like mesh distortion).

What is the meaning of "unbounded" ? I hear that Finite Volume is the lowest order of DG method (the basis function is constant), is it true!

The constant in cell approximation in finite volumes is the lowest order approximation. Of course you can obtain higher order approximations by various reconstruction techniques. But what are the advantages of DG ?