How many maritime launch platforms has ever been built and used for orbital space launches?
For dedicated orbital rocket launch platforms, there has only been a handful, Sea Launch being the most notable one, along with a Chinese one, and Italian Broglio System. Of these, only Sea Launch was truly mobile, the others were fixed platforms on the sea rather than ships.
Virtually every submarine that carries nuclear weapons can at least launch something sub-orbital, many have actually launched orbital satellites as well, as a means to get rid of a surplus of weapons post-cold war. In addition is any Aegis BMD system, of which there are many, which can launch weapons in to space to destroy incoming ballistic missiles. Getting an accurate count of all of these past systems would be difficult, but per Wikipedia there is around 30 active ballistic missile submarines and 36 Aegis equipped BMD ships, 33 US and 3 Japanese. Previously active systems are numbered even higher, but I'm probably not going to be able to get an accurate count of such systems.
Of these systems, I have found some evidence that Russia has launched satellites using some of these submarines, but the number of these launches I am having a difficult time getting a reasonable count. I can't find any evidence that the US has done so.
Undoubtedly not a complete list, but the "easy pickings" include Sea Launch
By 2014, it had assembled and launched thirty-two rockets, with an additional three failures and one partial failure. All commercial payloads have been communications satellites intended for geostationary transfer orbit with such customers as EchoStar, DirecTV, XM Satellite Radio, PanAmSat, and Thuraya.
The launcher and its payload are assembled on a purpose-built ship Sea Launch Commander in Long Beach, California, US. The assembled spacecraft is then positioned on top of the self-propelled platform Ocean Odyssey and moved to the equatorial Pacific Ocean for launch, with the Sea Launch Commander serving as command center. The sea-based launch system means the rockets can be fired from the optimal position on Earth's surface, considerably increasing payload capacity and reducing launch costs compared to land-based systems
and a Chinese platform.
It appears the Sea Launch platform is now owned by a Russian conglomerate, perhaps more for 'show and pride' than for actual functionality.
Thanks to "OM" for mentioning an Italian platform.
Developed in the 1960s through a partnership between the Sapienza University of Rome's Aerospace Research Centre and NASA, the BSC served as a spaceport for the launch of both Italian and international satellites (1967–1988). The center comprises a main offshore launch site, known as the San Marco platform, as well as two secondary control platforms and a communications ground station on the mainland.
In 2003 a legislative decree handed the Italian Space Agency management of the center, beginning in 2004, and the name changed from the previous San Marco Equatorial Range.2 While the ground station is still in use for satellite communications, the BSC is not currently used as a launch site