I wonder if it is possible to reproduce a transformer that could run those series incandescent lamps. I don't know much about series streetlighting, so I don't know what kind of transformer series incandescents use, but is it theoretically possible to build my own?

Hey Vince, I believe the old series streetlighting circuits operated off a special transformer (Constant Current Regulator?) that ensured a constant current was fed to the circuit at all times. The current used varied from system to system but the ones that I could remember off the top of my head were 6.6, 15, and 20 ampere systems. I loosely recall that the lamps used were rated by lumens and current instead of wattage and voltage. I also think the lamps ran at quite a low voltage compared to regular household incandescents but I don't remember offhand what was the voltage. It might be possible to build a small series regulator but I myself don't know exactly how to. Maybe someone here more knowledgeable can help you with that. I read that series lighting circuits are still commonly used for airfield and runway lighting systems so you should still be able to find some specifications online.     

I know that lighting collector Joe Maurath has been running a single 6.6A street series lamp off a Mailbu 300w, 12v low-voltage lighting transformer with good results. It would be much harder to find a transformer to run a 20A lamp, and you'd likely have to run it off a 240v mains circuit.

You can also still buy a constant current regulator but that might be a little overkill!

Finally, a friend of mine used a multiple lamp F32T8 electronic instant-start fluorescent ballast to create a small series loop using small incandescent lamps. I've thought about wiring my landscape lighting in series, but troubleshooting burned-out bulbs would probably drive me crazy and it just isn't practical.

The "consant current" supply unit is nothing more, then a transformer with many output voltage taps, selected by some sequencial selector. This selector is controlled automatically by the current sensing feedback: Current less then threshold => switch to higher voltage tap and vice versa. The selector usually uses multiple stages: Raw, finer, finest,... (e.g. on 400V maximum output are on one end 10 taps spaced 10V each, on the other end 10 taps spaced by 1V, so finally the system might set anything from 301 to 400V in 1V steps)

There's no need for current regulation on an incandescent lamp. Voltage is all you need to know, but if you don't... I, myself, would start at a low voltage, and increase until the lamp draws the rated current. There might be a better way to do this with a resistance reading, but I am not versed in that sort of thing.

 Yep, i was describing, how the street light supply unit work. There it has to compensate for "unknown" drops like wiring, burned out lamps (so bypassed), etc.
For lighting only one lamp for experiments this would be, of course, too complex gear.
determine the voltage as above, but then start the lamp first via resistor and only when it heat up (~0.5..1sec) connect it directly, to avoid current shock on cold filament.

For a 20A you might try looking at automotive battery chargers.  Depending on the type, they have a current selector switch.  I have a small basic one with selections of 1, 10 and 50 amps, but there are models out there with a 20 amp option too.