Addit: I am still really worried about components blowing up. I've just seen a patient who had a lead acid battery blow up in his face, and while electronics may be smaller, usually you have your face a lot closer too. Stuff blowing up is dangerous.
I've been pondering the schematic more. There are still a lot of things that need fixing, but for people (me included) who prefer to try things rather than draw them up on paper, the safe option is to use three cheap batteries to power this circuit. The cheaper the better, because cheap zinc batteries have a higher internal resistance and so are less likely to be able to deliver currents that can blow things up. I believe this is what manuka recommends and there is a good reason for choosing cheap zinc batteries over, say, alkalines or rechargeables.
The nice thing about the 4000 CMOS range is the wide working voltage eg 3-15V so picaxe and 4026 and 3 batteries would be perfect.
Re the 555 teaser, here is a little online calculator
http://www.coolcircuit.com/tools/ne555_calculator/index.php?R1=1000&R2=10000000&C1=0.1&T1=&T2=&sub=
The cap charges up via R1 and R2, and discharges via just R2. So if we make R2 much bigger than R1, the charge and discharge times become very close to being equal. There are a few tricks though. If we make R1 equal to zero, it shorts out the discharge transistor and this would cook the chip. So the current through the discharge transistor on leg 7 needs to be kept low - eg 5mA, so this sets R1 at 1k.
R2 then has to be much bigger. Let's try 100k. Plug in 10uF into the calculator and we have a duty cycle of 50.25%. We can do better than that though. Let's try 1000 and 10,000,000 ohms. Now it is 50.0025%. That is pretty close to square.
But if you use big resistors for R2, this does mean that the capacitor needs to be fairly small. For longer time delays eg minutes, the next trap is that the self discharge of the capacitor can be an issue, so you need to use low leakage caps like tantalums or polystyrene or greencaps. The capacitor leakage would affect the duty cycle more than the 1k when the resistor is more than 100k or so.
If you really wanted 50.000% duty cycle, run it through a divide by 2 chip that clocks on the leading edge. But for 49% to 51%, a 1k, a 100k to 10M and a suitable cap will do the job.
Good old 555 and picaxe can go together in a very handy way. Say you want to turn on a pump, but you want to be absolutely certain that the pump was never turned on and off many times a second. There is a very small but not zero chance that a haywire picaxe with lots of nearby RF hash or mains hash (eg flouro tubes turning on) could fail in that way. But a 555 is very tolerant of power supply noise and hash, so use a picaxe to trigger a 555 monostable and use a 555 to drive the pump.
Oh, and steirny's square wave circuit is even better!