A piston steam engine, for a pogo stick, is likely rather technically simpler than an internal-combustion-engine pogo stick. You need a water tank, a "flash" boiler (link), a steam-condenser, and a water pump (for moving water from the tank into the steam generator, against the steam pressure).
I think a quite-small-diameter piston-powered water pump, and a simple connecting rod between it and the main pogo-stick piston-shaft, can be workable. More complicated (but still necessarily mechanical, for this Idea to qualify as "steam-punk") will be the control system for the main power cylinder valves.
The best thing about this Idea is that the rate at which steam needs to be generated is much lower than in any other type of steam engine, simply because the piston cycles so slowly in normal pogo-stick operation. This should make it quite easy to condense the steam as fast as it is generated (normally it is very difficult to completely recycle the water used in a steam engine, because there is too much waste heat to dump).
It should be noted that as the water is recycled, the temperature of the water in the water tank should rise to almost --but not quite-- the boiling point. There is no reason to try to cool the condensed steam any more, than a little below the boiling point, after it condenses from vapor to liquid.
In fact, what we would like to have happen, to the extent possible, is for some of the steam to condense inside the power cylinder near the end of the power stroke. Use of the "heat of vaporization" (linked) qualifies as a loophole in how the "Carnot Cycle" (linked) specifies the efficiency of a heat-engine (allows the engine to be MORE efficient). Only the steam that doesn't condense needs some cooling to become also-condensed.
Well, that's all.-- Vernon, Dec 20 2011 Flash boiler http://en.wikipedia.org/wiki/Flash_boilerAs mentioned in the main text [Vernon, Dec 20 2011] Steam engines and the heat of vaporization http://books.google...QTP10M_xegkCo&hl=enAs mentioned in the main text. [Vernon, Dec 20 2011] Carnot Cycle http://en.wikipedia.org/wiki/Carnot_cycleAs mentioned in the main text. [Vernon, Dec 20 2011] Completely bohemian... https://marketplace...-Pogo-Stick/2660263 [2 fries shy of a happy meal, Dec 22 2011] Providing it is made of ornately decorated cast iron, with polished brass and copper fittings, french polished mahogany handles and a large centifugal governor on the top, we will award a bun.-- 8th of 7, Dec 20 2011 //More complicated (but still necessarily mechanical, for this Idea to qualify as "steam-punk") will be the control system for the main power cylinder valves.// Not at all. A bump valve for steam injection, and exhaust ports in the cylinder wall, makes control trivially easy; timing becomes a non-issue.-- spidermother, Dec 20 2011 [+]-- FlyingToaster, Dec 21 2011 [+]-- Klaatu, Dec 21 2011 yes and a bun for your last sentence! +-- xandram, Dec 21 2011 Hurrah for broken scolded bones!-- saedi, Dec 22 2011 Scalded too!-- spidermother, Dec 22 2011 I'd swear I've actually seen something like this before (not a pogo stick per sé, but a steam-powered bouncing thing). If I remember what the hell it was I'll post a link, but I invite others to join me in the search, because as all who were raised by wolves know, the hunt is the fun part.-- Alterother, Dec 22 2011 (+) If you listen closely you can hear him bark at the moon.[link]-- 2 fries shy of a happy meal, Dec 22 2011 Is it anything like a steam pile driver? I hope so. Where do you carry the coal?-- harperolocito, Dec 23 2011 Yeah. What does the burning? I imagine that solid fuel would get knocked all over the shop. Maybe gas might be the best.-- wagster, Dec 23 2011 random, halfbakery