It was quite a simple set of stations! There was a little disruption on the Piccadilly Line plus an annoying occurrence with the platforms at Hammersmith. It was a live draw so give yourself a little time.
Camden Town
Hyde Park Corner
Borough
Royal Oak
Finsbury Park
West Brompton
London Bridge
Piccadilly Circus
Chalk Farm
Tower Hill
Pimlico
Covent Garden
Angel
Hammersmith (H&C)
Bond Street
My route was:
Camden Town

Chalk Farm

Euston

Finsbury Park

King's Cross St Pancras (via Victoria Line)

Bank

Monument

Tower Hill

Monument

Bank

Stockwell

Green Park

Bond Street

Holborn

Earl's Court

West Brompton

Earl's Court

Hammersmith

Royal Oak
I teamed up with the Rail Riders for this challenge (they're the really organised team of kids from Sussex).
The crowds at Camden Town were about as large as I predicted, so it was a nice sedate shuffle down the spiral staircase towards the Edgware branch northbound platform. At the bottom, we took the chance to speed up, and about half of us (including me + team) piled on board a waiting train. We took this towards Chalk Farm, where there was a short doubleback. The train was a Charing Cross branch one, so not ideal, but we charged off the train, round the corridors and down to the Victoria Line, where we were blessed with a second waiting train. We all boarded it and were whisked up to Finsbury Park.
At Finsbury Park, I went up the stairs, but almost headed back down to the same platforms again! My teammates set me straight and we headed to the southbound platforms; no time was lost here though. We climbed aboard a southbound Victoria Line train, which we took to King's Cross St Pancras, where we had a 1 minute wait for a Northern Line train south. Down to Bank, up the escalator link and onto a waiting District Line train (the third train right there so far) to Tower Hill. Tower Hill was the first real wait of the day, as a District Line train was missed by seconds. A train turned up around 5 minutes later and we were off to Monument again, where it was straight down to the platforms and onto the Northern Line southbound. Tangy, who had missed the Northern Line at Camden Town, caught up with the pack at this point.
At Stockwell, we charged over the footbridge, down the Northbound Northern Line platform and to the Victoria Line. I reached the waiting train a few seconds before it left, but my teammates weren't anywhere in sight. As it is unwise to hold the doors for an excessive period of time, I allowed the train to go as a sprinting Tangy arrived on the platform. Thankfully, this being the Victoria Line, a train was there within another minute (enough time to let us get a decent door position), and we were off to Green Park. At Victoria, an entire country was waiting on the platform to board the train, but we were able to keep our position and it was a fast change onto a waiting Jubilee Line train at Bond Street, followed by an equally speedy change onto a waiting Central Line train at Holborn (the fifth such train, or sixth if you count Stockwell).
At Holborn, we were caught up in the crush of people on their way to the Piccadilly Line (a relic of the earlier disruption on the line). Luckily, at the bottom of the escalators is a little shortcut onto the westbound Piccadilly Line, which we were able to use, and this probably got us up a train as it was arriving as we reached the platform. We sat on the Piccadilly all the way down to Earl's Court, where it was a few minutes wait for a Wimbledon train. At West Brompton, there was a seventh/eighth waiting train taking us back to Earl's Court, where we changed onto a waiting District Line train towards Richmond (number 8/9!). There was no sign of the others now, so we'd left the pack behind. At Barons Court. we pulled alongside a Piccadilly Line train and unwisely bailed onto it (although the others were on board, this was in error as the District Line train overtook it on the stretch towards Hammersmith). We left the train at Hammersmith for the run to Hammersmith (H&C), where IT happened.
Upon arrival at Hammersmith (H&C), there was one train in the platforms (in Platform 3). The departures board said "First Train Platform 1"; I believed this to be due to a recently departed train on the platform as it was empty, so we all boarded the train. Then we watched as another train arrived in Platform 1. Now uneasy, I decided to keep an eye on the signal for the Platform 1 train in case it really would go first. Slightly over a minute later, the signal changed, but it was no use because the train left straight away. The train on Platform 3 did leave next, but once we'd confirmed that Peter was not on the train, we knew he'd reached Royal Oak first, and sure enough, there he was. We stopped the clock there.
All in all though, 9 trains in platform is an astonishing count for a system with certain unreliability!