Multi-round Transposition/Playfair as a secure hand cipher in the computer age Given that computers can be easily compromised, there is a need for a hand cipher that can withstand computer attack.  Clearly the traditional hand ciphers fail.

One time pads work.  They also have the important property of being able to burn used keys and thus prevent decrypting old messages if keys are revealed.  But key management is obviously hard.

Schneier's Solitaire cipher is certainly secure.  But it is also completely impractical as a hand cipher.  One would need to be extremely diligent to correctly encipher even a very short message without making a mistake that would make the rest of the message undecipherable.

So strengthening the traditional hand ciphers seems to be a more fruitful approach.  Of the many traditional ciphers, Transposition/Playfair seem to go together beautifully because Playfair combines two letters together and transposition then scatters them.  Transposition defeats the traditional methods of breaking Playfair, and Playfair defeats anagraming transpositions.

The idea is to take the plain text, transpose it, Playfair the result, then transpose that, Playfair that result again.  After two rounds each ciphertext letter is distributed to four plain text ones.  After three rounds eight.  

Both Playfair and Transpositions are relatively easy to do by hand.  In particular, Playfair processes two letters at a time, which is faster than ciphers like one time pads.

Transform/Playfair would certainly be secure if enough rounds are used.  It becomes very hard to analyze the intermediate values.  Each Transpose and Playfair would be performed with a different key, possibly based on some key schedule, so the key space would be huge.

But the question is, how many rounds are needed?  My guess is three, but that is just a guess.  

A crude brute force attack will certainly not work, there are 25!^2 combinations in each round.  There would be very little to hill climb on.  But what would be a more effective cryptoanalysis?  

Or are their better approaches?

("Secure" means in the context of hand ciphers.  So no chosen plain text attacks.  There will be a relatively small amount of text encoded, certainly not megabytes.  Etc.)

(There are other threads about hand ciphers, but I want to concentrate on the Transpose/Playfair here.)
Some years ago, someone (Terry Ritter maybe) suggested using Playfair as follows. 

First: just normal Playfair with some key.
Second: add a single null at the front and at the end of the cypher text.
Third: go to First with new key.

This can be extended by repeating the process. The point of adding the nulls is to split letter pairs between rounds. You could improve it as suggested in your question by using a transposition cipher before the Playfair step. Computer attacks on Playfair often just exhaustive key guesses coupled with Playfair's lack of ability to diffuse the key through the plaintext. Transposition breaks up the keyword search and adding single null between steps increases diffusion.

Still, one needs a fairly long key for the Playfair part. Perhaps using parts of a shared book keyed off the date. (Bible, "Catcher in the Rye," "Texas Almanac," etc.)