When Paul Gordan became a professor in 1875 he could show the binary form in any degree has some finite complete system of (general linear) invariants, but he could not actually give a complete system above degree 6. He discussed this limitation that year in Uber das Formensystem binaerer Formen (B.G. Tuebner, Leipzig).

So far as I can tell (by reading Kung, Sturmfels, Derksen, and Eisenbud and some correspondence with them) advances in theoretical and computer algebra have not really changed that. The complexity of calculation rises so quickly with degree that the limit of degree 6 has not been passed by much if at all. But perhaps my information is incomplete or out of date.

Is it now possible to calculate specific complete systems of invariants in higher degrees? What is the state of that?

Answers to the question Algorithms in Invariant Theory give relevant references but they do not give any clear answer. One leads to an arXiv article which describes one computer package this way.

The package calculate the set of irreducible invariants up to degree min(18, βd), but in all known computable cases this set coincides with a minimal generating set, see, for example, Brouwer’s webpage http://www.win.tue.nl/~aeb/math/invar/invarm.html

That refers to the degree of the invariants, not the degree of the form they are invariant for. I did not find a description there of which cases are computable. And that link no longer works.