We present the first high-statistics study of the etaeta system over t
he mass range 1000-3000 MeV/c2. The experiment was performed at the Fe
rmilab Antiproton Accumulator, and the data sample consists of six-pho
ton final states produced in antiproton-proton annihilations at square
-root s in the range 2950-3620 MeV. We find evidence for three states
with masses 1488 +/- 10 MeV/c2, 1748 +/- 10 MeV/c2 and 2104 +/- 20 MeV
/c2 respectively.