The first Action Comics Annual published by DC was released in 1987 in the post-Crisis era.
However, Federal Comics beat DC to the punch when they published Action Comics Annual #1 a few years earlier c.August 1984:
This issue was based on Action Comics #544, the 45th Anniversary issue:
The two stories in this issue - Luthor Unleashed! and Rebirth! - are the first two features in Action Comics Annual #1. The Federal edition printings were modified in various ways.
This is page 1 of Luthor Unleashed! in Action Comics Annual #1:
The art has been extended in each panel at the bottom of the page - compare to the original page in Action Comics #544:
This is the fifth page of Rebirth! in Action Comics Annual #1:
However, in the original story, this is the final page of the story:
It includes the text box advertising the next issue, which was excised in the Federal printing, but I cannot explain why the sequencing of the pages was adjusted.
Action Comics #544 also contains a few extra features: A three-page text feature titled Happy 45th Anniversary, Superman! by Jerry Siegel; A Short Message from Joe Shuster; a full-page colour pinup by Shuster dated 1983; and two single-page pinups of The New Luthor and The New Brainiac. Only the Luthor pinup is included in the Federal edition. As noted on AusReprints, the credits associated with these absent features still adorned the Federal edition cover.
The next story in Action Comics Annual #1 is ...With But A Single Step! from Action Comics #545. This is a 23-page story presented in 22 pages in the Federal edition. This is achieved by reducing and reformatting the following two pages:
This is the Federal version:
Ironically the art in the second panel needed to be extended in order to be reformatted as a standalone panel from the original inset panel.
The final page of the story was also amended by removing the next issue details:
At least this made sense as the Federal issue contains the next story Showdown! from Action Comics #546. This is also a 23-page story presented in 22 pages. Here's the original pages 18-19:
This is how these two pages are condensed into one page in the Federal edition:
There's some radical surgery on this page: panels are not only reduced but they are variously cropped, reformatted on the page, reshaped and figures are omitted.
The final page of Showdown! includes a next issue blurb:
This is omitted in the Federal edition:
The two-part Planeteer story advertised at the end of Showdown! appeared in Superman #387 and Action Comics #547. Federal Comics reprinted at least one of these instalments a month later in Action Comics #6. I haven't spotted the other chapter in a Federal edition yet, and if it doesn't exist, that may partly explain why the Federal edition didn't take the opportunity to promote a 'next issue' - and unfortunately, there would not be an Action Comics Annual #2.
Interestingly, Action Comics Annual #1 was advertised in Federal Comics' Action Comics #5 c.July 1984:
This blurb appears at the end of the story Within These Hands--Power!, originally published in Action Comics #543.
So, we can be certain that Action Comics Annual #1 was published in between Action Comics #'s 5 and 6.
Federal Comics' The New Teen Titans and Supergirl series were advertised on the inner covers