Upper and lower estimates of eigenvalues of the Laplacian on a metric graph have been established in 2017 by Berkolaiko, Kennedy, Kurasov, and Mugnolo. Both these estimates can be achieved at the same time only by highly degenerate eigenvalues which we call maximally degenerate. By comparison with the maximal eigenvalue multiplicity proved by Kac and Pivovarchik in 2011, we characterize the graphs exhibiting maximally degenerate eigenvalues which are the figure-of-eight graph, the 3-watermelon graph, and the lasso trees-namely, trees decorated with lasso graphs.