American Top 40: The second program of the original 1970-1995 series – aired July 11, 1970 – is thought to be lost, although later, a "July 11, 1970" program was supposedly created featuring segments from shows aired in July and early August, 1970.
Different stories exist as to the status of Casey Kasem's 1990s countdown program, Casey's Top 40, which Kasem created following his August 1988 departure from AT40. Casey's Top 40 (which aired from January 1989-March 1998) was distributed by Westwood One Radio Networks, and it is unknown whether the shows exist, much less in playable form. As of the current time, Premiere Radio Networks (which distributes reruns of AT40 programs from 1970-1988 to radio stations) has not acquired rights to that show. (Premiere does own rights to AT40's current version, which began in March 1998, hosted by Kasem through January 2004 and Ryan Seacrest since then.)
Although all programs from the Shadoe Stevens-era (1988-1995) exist and Premiere owns the rights, they are not currently airing. It is not known when or if Premiere will re-air those programs, either as part of the 1980s package (for 1988-1989 programs) or as a separate package.
All episodes from AT40's sister program, American Country Countdown, are believed to exist in its entirety from the show's beginning (since 1973), but – aside from the trading circuit or various download services – they have almost never been replayed. Those that have probably are the occasional specials – as opposed to regular weekly programs – and usually on small, rural stations. Although the shows have been remastered by Charis Music Group (the folks that remastered AT40's entire run), it is unknown who owns the rights to possible future repeat airings.
Tales by the Fireside, a popular BBC world service radio programme of the narrator's (Lionel Marson M.C.) experiences as a soldier during both World War II and the Great War. Despite being effectively the voice of the BBC's world service, and thus Britain, during his time with the station, no copies of the programme are known to have survived to this day. Even the name of Lionel Marson is less than well known, due to his being active on air at a time when mentioning one's name while broadcasting was considered extremely poor form. At the time, he was simply recognised by his voice. This tradition was first broken only shortly before he retired.
The current generations of his family are actively searching for recordings.
Many of The Firesign Theatre's live radio performances, such as their Hour Hour shows, contained music that was legal to broadcast live but means they can't be sold as aired. The official Firesign archivist is editing much of this material for just that reason.
Radio Mystery Theater.
Paul Harvey's News and Comment and The Rest of the Story shows. His website used to have some archives, but the site was taken down after he died. This is a good place to start.
Since 1986, The Bob & Tom Show has released albums that compile popular recent skits, guest appearances, songs, etc. at the rate of at least one per year, as well as Greatest Hits Album-style compilations for particularly popular characters or topics. All of them go out of print within a few years and their Unintentional Period Piece nature renders reissues out of the question (even more so for the collections made before the show went national in 1995, because they included many Indiana-specific skits/songs), so completist fans must go to Ebay to track the older ones down. Worse, two early titles (A Day at the Race! and Good Morning Saudi Arabia) were cassette-only releases.
Vivian Stanshall (late of The Bonzo Dog Doo Dah Band), made numerous BBC radio sessions for the John Peel Show, featuring comic monologues and songs, mostly based on his "Rawlinson End" concept. There have since been an album, a book and a film of Rawlinson End, but although the BBC has released lots of other "Peel Sessions" on CD, Stanshall's remain in the vaults apart from the odd rare broadcast and bootlegs.
The radio version of Another Time, Another Place only seems to be available through the academic database Box of Broadcasts, which can only be accessed via British universities.