a(3) is known to be 15 or 16. I have put 15 here because otherwise the sequence would be too short to include in the OEIS. But the true value is (or was in 1997 at least) not known.

REFERENCES

L. W. Beineke, Biplanar graphs: a survey, Computers Math. Applic., 34 (1997), 1-8.