Stages of the Apollo stack (at least the configuration that landed humans on the Moon) are identified as S-IC, S-II, and S-IVB for the first, second, and third stages. There's no S-III. Why the gap in numbering? Was Saturn going to be a four-stage rocket at some point in its development?
The planned scope of the Saturn launcher program varied quite a bit before the launchers were finally developed, as you can read here. The basic idea was to progressively develop a larger and larger rocket, re-using upper stage designs while enlarging the lower stages. The exact parameters of the design were altered repeatedly during early development, so the overall picture is complicated and confusing.
Five basic stages would be developed, S-I to S-V, with S-V being a modification of the Centaur stage, using 2 RL10 engines. These stages would be used in different 3- and 4-stage configurations. The S-III stage would have used 2 J-2 engines and S-IV would have 4 RL10s.
The idea was that the line would begin with a three-stager using a modestly sized S-I stage and the S-IV and S-V stages (eventually this became the two-stage Saturn I with a 6-engine S-IV stage) and the first stage would be progressively upgraded with more of the intermediate stages used. (S-V never happened; Wernher von Braun was never a fan of the Centaur, possibly because it was primarily developed for the Air Force, while von Braun's fortunes were tied to the Army until his transfer to NASA in 1960.)
However, the Kennedy challenge to land on the moon before the end of the 1960s accelerated the development of the program, and development of the S-III was skipped because the larger S-II stage was going to be needed sooner.
The S-III stage did exist, but only on paper. It was proposed for some Saturn-C configurations.
There was the original Saturn C-2 variant with the stages S-I, S-III, S-IV and even a S-V.
The original Saturn C-3 would have been a combination of S-I, S-II and S-III, as well with S-IV and S-V on top, which would have made it a five-stage rocket.
But in further development the S-III and the S-V have been dropped, leaving only S-I, S-II and S-IV.
The other answers are okay as far as they go but I prefer something less conventional.
There was an S-III stage and it was in fact the most flown Saturn stage. But for obscure reasons it was designated S-IVB instead of S-III and the vehicle on which it was first flown was designated Saturn C-IB (later Saturn IB) instead of Saturn C-2.
3$\begingroup$ Do you have any references so that readers can check your assertions for themselves? I, for one, would like to hear the "obscure reasons", otherwise this answer boils down to "because, reasons". $\endgroup$ Oct 4, 2018 at 21:11
$\begingroup$ @OrganicMarble How to hook a new user better? Down votes should not be allowed on new users.+1. Is the answer incorrect? $\endgroup$– MuzeOct 5, 2018 at 4:05
3$\begingroup$ I wasn't the down voter. Don;t make assumptions. $\endgroup$ Oct 5, 2018 at 4:39
1$\begingroup$ @jimDavis, I think your question is good, and maybe OrganicMarble should have behaved more friendly, but he has right that giving references significantly improves the acceptance of your posts. $\endgroup$– peterhOct 6, 2018 at 17:07
2^xsequence. Waiting for Saturn 65536 $\endgroup$