Does anyone know of any places in the world that still have installations of 1880s to 1910s carbon arc street lanterns that have NEVER been retrofitted and are still in active use?

That's virtually impossible. In the late years of arc lighting (late 1910s to mid 1920s) there were "metallic flame" and magnetite arc lamps whose electrodes could last around 150 to 175 hours before needing to be trimmed. But they probably weren't that popular because their design restricted them to DC operation, and AC was already the standard for power distribution. And even then, arc lamps could run without maintenance for a week at best.

Arc lamps made it to 1920 only because high-power incandescent lamps took a fairly long time to be perfected, and before that time arc lamps were the most efficient way to provide tens of thousands of lumens. It's the Mazda 'C' lamp that changed everything. Being gas-filled instead of vacuum (as were the previous 'B' lamps), they could run much more efficiently, and were the first suitable replacement for arc lamps.

That's virtually impossible. In the late years of arc lighting (late 1910s to mid 1920s) there were "metallic flame" and magnetite arc lamps whose electrodes could last around 150 to 175 hours before needing to be trimmed. But they probably weren't that popular because their design restricted them to DC operation, and AC was already the standard for power distribution. And even then, arc lamps could run without maintenance for a week at best.

Arc lamps made it to 1920 only because high-power incandescent lamps took a fairly long time to be perfected, and before that time arc lamps were the most efficient way to provide tens of thousands of lumens. It's the Mazda 'C' lamp that changed everything. Being gas-filled instead of vacuum (as were the previous 'B' lamps), they could run much more efficiently, and were the first suitable replacement for arc lamps.

I wonder when the very last carbon arc street lanterns in the world were taken out of active service.

An ad in the April 1940 edition of the American City Magazine is interesting. General Electric details how the City of Niagara Falls “ Had one of the finest luminous arc street lighting systems, yet it recently relighted 23 miles of street for better night traffic safety. More than 1000 arc lamps were replaced with modern efficient incandescent luminaires”. There is a photo of the GE Form 81 luminaire. So arc lighting existed into the 1940’s.

I'd imagine the parts were probably made until the 1980s. According to here the last carbon arc searchlight was manufactured until 1981.

Years after incandescent and HID replaced carbon arc for general lighting they were still used for specialist applications such as searchlights and film projectors. The xenon short arc lamps superseded carbon arc in those applications though.

Anyway Toronto lit its last carbon arc roadway light on October 31st 1911. After than they switched to low mounted incandescent luminaires. Only around the 1940s did they switch back to high mounted incandescent luminaires that were over the roadway. 

An ad in the April 1940 edition of the American City Magazine is interesting. General Electric details how the City of Niagara Falls “ Had one of the finest luminous arc street lighting systems, yet it recently relighted 23 miles of street for better night traffic safety. More than 1000 arc lamps were replaced with modern efficient incandescent luminaires”. There is a photo of the GE Form 81 luminaire. So arc lighting existed into the 1940’s.

Providence, RI had hundreds of post top lanterns powered by luminous arcs up until at least 1940 and/or right after WW2. These had ornamental milk-glass globes. Their successors were Union Metal (?) upright poles with curved arms and matching under-supports beneath. These had either incandescent or 400W or 250w(?) MV lamps with their round can ballast and glass tube-powered Fisher-Pierce photocontrol mounted on top of each of the poles. Many of these old MVs survived through the late 1980s when HPS invaded.

Well, that's kind of odd, honestly!

I mean, the difference in terms of maintenance requirements between even the most advanced arc lamps and the earliest incandescent fixtures is pretty significant. We're talking about what, 175 hrs vs. 1 000 hrs? Not quite an order of magnitude, but still a big step forward! And I can imagine it was much quicker to replace an incandescent lamp than trimming an arc lamp, which seemed to be an art in and of itself!

Yeah the reduction in maintenance required between a luminious arc system and a incandescent system must have been revolutionary for streetlighting. I guess the same could be said when the incandescent systems were replaced with MV, epsecially the later MV lamps.

Anyway I suppose another reason was that carbon arc luminaires were rather complex, more like HID luminaires in the amount of gear inside. You needed some sort of control gear to run the arc as I believe most arc lights ran off of DC instead of AC. And then you needed some means to advance the carbon rods in order to maintain an constant arc gap. I believe some rather elaborate systems involving solenoids were used during the tail end of carbon arc lighting.

On the other hand incandescent luminaires really only required a suitable lampholder inside. The multiple lights could be hooked directly to the mains. The series systems needed a constant current regulator but one of those could feed a long string of lights. 

The original arc lamps were indeed designed for series DC circuits, and most used a set of two solenoids (one in series with the arc and the other in parallel) to operate a "ring clutch" which lifted the upper carbon to strike the arc then adjusted the gap as the electrodes burned away. Parallel/multiple arc lamps are actually simpler because only the series solenoid is needed.

The switch to AC didn't really make arc lamps more complicated either. They were using inductive ballasts instead of resistors.

By around 1920 the arc lamp mechanism had been extensively simplified. But yeah it was still very much more complex than any incandescent fixture!