The Saturn V had an initial production run of 15 vehicles (plus 3 test articles). Twelve were used for Apollo 4, 6, and 8-17, one was used for Skylab, the components of the other two are on display in various locations. I've heard there was originally going to be a second production run of the Saturn V, probably including the uprated F-1A engines among other improvements. Was this ever officially considered? If it was, when was it cancelled?
Russell is correct; no second run was ever officially ordered, which is why we never saw more Saturn Vs lift off. However, when you ask if it was ever considered, know that it certainly was - in a big way! I suggest you look into Saturn MLV and ELV studies...
MLV stood for 'Modified Launch Vehicle' while ELV stood for 'Evolved (or Earth) Launch Vehicle.' These launchers would have been improved Saturns with greater payload capacity, better economy, or both.
Saturns SA-501 through SA-512 were all three-stage Saturn Vs used for Apollo missions, while SA-513 was a two-stage vehicle (not INT-21!) that launched Skylab. SA-514 and -515 were to launch another Apollo mission and a backup Skylab, respectively.
The 'second run' would begin with standard Saturn Vs marked SA-516 onwards, with perhaps some two-stage launches and the eventual introduction of INT-21. But the MLV variants would begin at SA-522 with the Saturn MLV-V-1A: stretched stages, six F-1s and seven S-II J-2s. SA-526 would introduce the first of the F-1A powered Saturns, the MLV-V-1, which would eventually use J-2S and J-2T. SA-536 would see the MLV-V-3 launcher with the high-pressure HG-3 upper stage engine, and later evolve into the V-3B with toroidal upper stage engines. These later variants would approach 170t to LEO and 75t to TLI.
Between these milestones, it is likely that solid booster assisted Saturns such as the MLV-V-4(S) would arise. 120in and 156in strap ons could boost payloads to around 160 tonnes.
After this, some studies get crazy. Launchers such as the ELV models of the Saturn V-25(S)U would use 260in diameter solid boosters and stretched stages to lift 250 tonne NERVA stages to LEO for mars exploration. The V-4/260 would also use 260in solids, but with massive 23m fairings and auxiliary tanks to lift over 360 tonnes. My favourite design involves a vehicle consisting of four stretched Saturn Vs joined under a single fairing to allow payloads of 520 tonnes and over!
Many, is not all, of these improved designs would likely never be intended to fly. A true second run would probably only see more standard Saturn Vs, joined by a few INT-21s, MLV-V-1s and various solid-boosted models. Sometime during the production run the rockets would no longer feature fins.
But of course, consideration does not mean real, in-depth planning. It seems the second run was canned around 1968, when Apollos 18 to 20 were also scrapped. Much of this appears to be designer dreaming without a goal to be achieved before the next decade was out...
What is true is that there were no orders for a second batch of Saturn V Rockets.
There were some attempts to raise budget for further Launchers in the budget plans for 67, 68 and 69. But all were denied by the Congress.
The denial of two further rockets, that should be produced in 1969, by the lawmakers on August 1, 1968 let to the end of the Saturn V production and in the end to the resignation of James Webb as NASA Administrator.
Nevertheless, the Saturn V production line was not closed then and in the integrated program plan (IPP) that Tom Pain and George Muller developed in 1969, the Saturn V played a vital role.
It was only in March 1970, that Tom Pain closed the production definitely, in somewhat that could be described as a "panic" reaction on the 20% cut that the White House made on the 1971 NASA Budget request.