According to Wookiepedia (backed up as well by my recollection of reading nearly everything EU had on the period between 0ABY and Vong):
One Death Star was completely built by the Empire, with a second and third one never reaching full completion, as well as a prototype being in existence. In addition, a scaled-down version, the Tarkin, and a Hutt knockoff, the Darksaber, would be created by the Empire and Durga the Hutt, respectively.
There's also a very popular fan theory that even DS2 wasn't meant to be "for realz" - and merely a convinient, albiet super expensive, "it's a TRAP" for the Rebellion from the get-go.
As far as "how many were planned", no canon information exists. But since Tarkin - as a Grand Moff - basically "appropriated" one, we can make an informed guess that - given unlimited resoursing - one would be created for each of the oversectors controlled by a Grand Moff.
I haven't found canon reference for this yet, but Wookiepedia mentions that there were 20 of those initially ...
Under the Sector Governance Decree, twenty of these new Moffs were also given responsibility for supporting the Republic's twenty Sector Armies, and their military territories, generally matching the sphere of influence of their respective Sector Armies, were dubbed oversectors.
... but later the oversectors and Grand Moff assignments changed and there's no firm canon info on what the final # was.
Interestingly, the main thrust of your question (how many were envisioned) was moot for the Emperor, who didn't view the Death Star as much a military vehicle as a symbol - and thus even one would have been enough for his purposes.
This is elaborated on in James Luceno's New Disney Canon novel "Tarkin":
... As powerful as the battle station might become, its real purpose was to serve as a tangible symbol and constant reminder of the power of the dark side, and to free Sidious from having to portray that part.