As Ivan commented, there are actually an infinite number of possible series of this type. So your question is really why there are only six named series. The reason is part of the culture of science. Typically, a result is named if it is, to use Mithoron's term, sufficiently notable. Thus we have the Diels-Alder reaction, the Hartree–Fock method, the Wentzel–Kramers–Brillouin (WKB) approximation, the Southern blot, and so on.
Hence the reason why the initial series of hydrogen spectra were named is because they were notable scientific results. [In addition—and perhaps someone more expert in the history of science can comment on this—it is possible that identifying the, say, fifth and sixth series weren't that scientifically notable, but a tradition had become established of naming successive spectra of hydrogen.] But beyond the sixth, finding more series was essentially deemed routine. The last published series I know of is the seventh, identified by Hansen and Strong, but not named after them [Peter Hansen and John Strong, "Seventh Series of Atomic Hydrogen," Appl. Opt. 12, 429-430 (1973)].
That's not to say it wouldn't be experimentally difficult to see higher-numbered series, since they become increasingly faint. Rather, it's that, at this point, finding them isn't interesting science.