About 21.3 km/s with 5 stages. To develop this model, I used Kerbal Space Program with Real Solar System & Realism Overhaul.
The rocket as built, by stage (dimensions given in diameter x length)
12 x 54 m stir-welded Al-Li semi-stringer construction tanks. 44 sea-level Raptors (330 s sl. Isp, 355 s vac. Isp). Total dry mass of stage: 257 tons. Total wet mass of stage: 4874 tons. Mass ratio: 18.9. Off-Pad SLT: 1.24. Burn time: 3.5 minutes. dV contribution: between 4.7-5.1 km/s.
12 x 15 m Al-Li integral construction tanks. 7x J-2X engines (448 s vac. Isp). Total dry mass of stage: 80 tons. Total wet mass of stage: 691 tons. Mass ratio: 8.6. Burn time: 5 minutes. dV contribution: 3.4 km/s.
8 x 15 m Al-Li integral construction tanks. 2x J-2X engines. Total dry mass of stage: 33 tons. Total wet mass of stage: 302 tons. Mass ratio: 9.2. Burn time: 7.5 minutes (1 minute burn required for orbital insertion). dV contribution: 4 km/s.
6 x 6 m Steel-Al-Li pressure-stabilized balloon tanks. 4x RL-10B-2 engines (465 s vac. Isp). Total dry mass of stage: 5.7 tons. Total wet mass of stage: 80.6 tons. Mass ratio: 14.1. Burn time: 13 minutes.
3 x 3 m Steel-Al-Li pressure-stabilized balloon tanks. 2x AJ10- Transtar engines (vac. Isp). Total dry mass of stage: 1 ton. Total wet mass of stage: 38 tons. Mass ratio: 38. Burn time: 38 minutes. dV contribution: 4.8 km/s.
Payload was slightly over 10 tons (10,010 kg), containing avionics, batteries, and lead ballast to make up the difference. All dry mass values for stages contain the estimated masses of decouplers, engines, turbopumps, fairings, etc. Payload fairings (which enclose the two balloon tank stages) decouple shortly after the second stage ignites. Hydrolox stages are either a Centaur-derived stage or use SOFI.
As you can see, mass ratios are rather south of pessimistic (likely because I used a generally more-than-optimal number engines per stage), except for the Transtar-derived upper stage. However, Transtar was in development to have an extremely low mass ratio (see reference), and I think with modern (Steel-Al-Li) balloon tankage, such a ratio is possible. The long burn time is also reasonable--the Transtar engine was derived from the Shuttle OMS, which had a EXTREMELY high rated burn time.
Burn time of the J-2X stages was kept to below the 500 s J-2 burn time. The RL-10B-2 stage was kept to below the 700 s burn time of the DCUS stage.
Together, this gives me about 21.3 km/s, 21.6 km/s if only vacuum dV numbers are summed. Again, note that my mass ratios are generally much lower than the that of the other answers'. If the middle stages also had ~20:1 ratios, then it is likely that you would achieve in excess of 23 km/s on the pad. Also, of course if you replaced some of those 10 tons of payload with some high-performance light solid stage, you would be able to get even more dV out of your setup. Tsiolkovsky is an inescapable god, and he loves more stages.
Additionally, I demonstrated that such a rocket could be flown to 200 km orbit successfully. Details may be found in this imgur album.
Such a rocket would look something like this upon liftoff:
And something like this after staging and fairing separation (forgive the unrealistic hydrolox vacuum plume, the RO dev team fixed this but I am still playing on an older test build):
If this KSP-RSSRO-sourced answer is not satisifactory for this SE, I will remove it.