Cassini: Data from the Imaging Science Subsystem's Wide-Angle and Narrow-Angle Cameras are converted to JPEG format and automatically contrast-stretched before release.
Curiosity: Data from Navcams, Hazcams, MAHLI, and ChemCam are available immediately; data from Mastcam and MARDI are available after a 24-hour delay. All are converted to JPEG format and automatically contrast-stretched before release. ChemCam images are also flat-fielded. Mastcam images are highly compressed, reducing their quality. Also available through Mike Howard's Midnight Planets and Joe Knapp's Curiosity Rover Images and Rover Image Synth.
Any Hubble Space Telescope images that are taken as a result of a request for Director's Discretionary Time are released immediately, in science-quality format. (I suggest reading this tutorial for guidance on using the Hubble data archive.)
I don't usually cover solar missions, but SOHO, SDO, and STEREO all share science-quality data immediately.
I'm also not familiar with all of the realtime Earth image resources, but notable ones include DSCOVR and Himawari for full-globe views. More detailed views from Terra, Aqua, and Suomi NPP are available via NASA's Worldview interface.
Missions that release image data in batches within weeks of receipt from space
New Horizons: Data from its Long-Range Reconnaissance Imager are released on Fridays, with each release including images returned as of the preceding Tuesday. Images are converted to JPEG format and automatically contrast-stretched before release. This mission is almost done returning all its data.
Juno: Data from JunoCam are released at weekly intervals, including data about a week to two weeks old, in a couple of formats. Right now and through October they are working on a "Marble Movie" with frames appearing at this buried location roughly once a week. When I notice that they have appeared, I download and repost them here.
Missions that do very rapid (within weeks to 3 months) release of science-quality data
(Apart from the ones mentioned above)
Rosetta: Navcam releases its science-quality data in monthly batches, with the most recent images being less than a month old. Sadly, that mission ends next week.