Calar Alto Observatory (CAHA)
For raw and reduced data from Calar Alto. CARMENES also has other data at the CARMENES-specific archive here.

Canada-France-Hawaii Telescope (CFHT)
Contains raw and reduced data from CFHT, as well as a number of other observatories, including Hubble.

ESO Phase 3 Archive
For reduced data from the European Southern Observatory.

ESO Main Data Archive
Contains a wider range of instruments, but features mainly raw, unreduced data. Data reduction tools and an FAQ page can be found here and here respectively.

Gemini Observatory Archive
For raw and reduced data products from the Gemini Observatory.

Gran Telescopia Canarias Archive
For raw and reduced data products of the GTC.

Italian Center for Astronomical Archives
Some data archives listed here (LBT, TNG) are run by IA2, as well as others, which are listed at the bottom of their main page.

Infrared Science Archive (NASA/IPAC)
Contains data from numerous telescopes, including but not limited to WISE, 2MASS, Spitzer, and the Planck Telescope.

Keck Observatory Archive
For raw and reduced data from the Keck Telescope. Reduced data must be explicitly requested in the interface above the file list, otherwise only the raw images will be available. A link to data reduction utilities can be found here.

Large Binocular Telescope Archive
Raw data from the Large Binocular Telescope. A link to data reduction utilities can be found here. There is also a separate link for LBT Interferometer data here.

NOIRLab Astro Data Archive
Contains data from Kitt Peak, Bok, Cerro Tololo, SOAR, and WIYN Observatories. The new NEID spectrograph at WIYN also has its own data archive here.

Subaru Telescope Archive (SMOKA)
Contains only raw data.

Telescopio Nazionale Galileo Archive
Note: Reduced products are interspersed with raw data with no explicit flag to tell the difference. For example, in GIANO-B spectra, the reduced, 1D spectrum files have names which end in "ms1d".