Using the same system of number-of-letters for numbers, one can get a little deeper into the number with a poem:
Pie.
3
"I wish I could determine Pi... Eureka!" cried the great inventor:
14159265358
"Christmas pudding, christmas pie is the problem's very center"
979323846
Anyone who feels they can compose a next line with word lengths 43383279 is welcome to do so. For further digits, check pi.
Personally, my favorite way of memorizing pi is to remember where the pi key is on my calculator. Any arithmetic involving more than a few digits pi should not be carried out by hand when there is a better way available. Unless your computer or calculator does not have sufficient precision for the task.
If, however, you are stranded on an alien world and you need to compute your ballistic course back to Earth, and your solution needs to have pi to better precision than 1 part in 1 sextillion... what are you supposed to do then, compute a sextillion terms of the alternating series approximation? No. You're not supposed to use that many digits of pi, because there's no way your rocket will be able to have that degree of precision anyway. Just figure out a way to make course corrections en-route. But assuming you have no choice...
... and supposing you do have a calculator capable of doing a square root and unlimited precision, spam the 2, square root, and + keys until you have twice as many nines in the result as you want digits of pi; then get the difference of this from 2, take the square root, and double the result until you get something looking like pi. Other series converge faster, but this is fast enough for most purposes, and easy to remember.
... and supposing your calculator instead has limited precision, it will probably have less precision than the digits given above, outright. Even if the calculator can hold 11, you will need to break up each number into multiple parts and carry out complicated carrying actions on the number each time you use it (especially nasty for those square roots). You might as well do it on paper. Either way, I hope you aren't error-prone, since whatever you are calculating probably is.