The paper

D.J. Broadhurst and D. Kreimer, Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Pade-Borel resummation, Physics Letters B 475.1 (2000): 63-70. (http://arxiv.org/pdf/hep-th/9912093.pdf)

presents QFT computations to 30 loops for Yukawa theory in 4D, which is probably impossible without the compact Hopf representation. Followup work by the same authors derives Schwinger-Dyson equations http://arxiv.org/pdf/hep-th/0012146.pdf to 500 loops for the same QFT.

Applications to resummed QED are given in http://arxiv.org/pdf/0805.0826.pdf (based on http://arxiv.org/pdf/hep-th/0605096) . Here most interesting is the fact that some of the resulting nonperturbative approximations do not have a Landau pole. The existence of the Landau pole in low order perturbation theory is frequently used as *the *argument to claim that QED probably cannot exist as a fundamental theory but only as an effective field theory. Thus the fact that (some of) the nonperturbative results here don't give a Landau pole shows that the problem of the existence of QED is in fact wide open.

http://arxiv.org/pdf/0906.1754 contains applications to QCD.

Note that everywhere in QFT one must make approximations, and the nature of approximations depends on the method used. In the epsilon-expansion one only considers a fixed number of loops and neglects everything else. The Hopf algebra approach accommodates many loop orders but neglects complicated diagrams. Which neglect is worse is not easily answered. To estimate the effect of neglecting something one must make two calculations, one neglecting more, one less, and the difference in the results gives an indication of the accuracy.

In some cases it is known that only fairly simple diagrams are relevant. For example, ladder diagrams are the only one's relevant when a nonrelativistic approximation is adequate, and renormalization amounts to resumming all repetitive diagrams into one sum. The resumming achievable through Hopf techniques amounts to something similar, hence it is quite likely an improvement compared to RG improved perturbation theory. But I don't know whether systematic studies of the errors have been made.