Double-peaked Balmer lines have been observed in about 150 active galactic nuclei and were interpreted preferentially as emission from relativistic accretion disks. We report the discovery of extreme double-peaked lines in SDSS J0942+0900 and SDSS J1417+6141. The FWHM of the Hα line, 40,600 km s-1, in the first object is almost twice as large as the broadest previously known. By comparing the line profile with accretion disk models, we find that most of the line flux is emitted from a narrow annulus around 100rg in SDSS J0942+0900, and from a disk with radii between 100rg and 400rg in SDSS J1417+6141. This is the first time that an accretion disk at radii below 100rg has been directly probed through optical emission lines. A certain asymmetry in the disk is required for both objects. Another, much weaker, broad Hα component (Wλ ~ 20 A and FWHM ~ 4000 km s-1) is also detected in both objects. Both objects show weak radio emission and strong high-ionization narrow lines.