Ok, you've attacked my choice of words, now attack my idea. Does it have greater entropy than his? You can say no and I'll take my toys and skulk away. It's that simple.
It seems you've described a few different things, which might be why you've gotten different answers about how much entropy your method has.
If you just remove double letters (or move them in a deterministic way, like sticking them on the end of that same word), then no, yours doesn't have more entropy than his, you just have a different list of words than he does. Which is exactly what I said in this post
If you tape the string "correcthorsebatterystaple" to itself as a loop and start somewhere else than at the first 'c', so for example might use "rsebatterystaplecorrectho" then you've added a factor of 25. Which is exactly what I said in this post
. As I then said, and as others have now said since then, this is indeed more entropy (4.64 whole bits of it), but is more convoluted and weird than simply adding a single random letter to the end of "correcthorsebatterystaple", such as ""correcthorsebatterystapleq" or "correcthorsebatterystaplej", which adds a slightly higher factor of 26 (4.70 bits of entropy).
Numbers don't really do anything for me in passwords. Pick your poison. I'm looking at what it is that I want to do. If he gets in my house he'll find mine written down. What I want is ease of implementation and secure storage with two keys. I want to hide them in plain sight. If he attacks the hash I want him to look at all the possible combinations for each character. Since my set is Randall's set do my changes increase or decrease the entropy? Am I making it harder or easier.
I have a secondary goal which fits me. I want to have my passwords available in meat space and on my network. I can hide them on my computer, I want to hide them on my desk. I keep a book on my desk. I create and index of phrases, from that book, as passwords. I design my index so that it picks the phrase in a quasi random way. Given that index and a mask to isolate the phrase I can read the password with the mask. Since it's a book, I could also hide it on my Kindle. I can do this with Randall's scheme.
The problem with increasingly convoluted methods is that they become increasingly difficult to evaluate. (And impossible when you describe it as vaguely as the above.) Better to have a simple method with a guaranteed minimum level of security than an obfuscated method that might turn out to be incredibly insecure.
If "all the possible combinations for each character" means you put the repeated letters at random places in the string, then there are 24 places to put the first one and 25 to put the second one, so you increase it by a factor of 24*25=600. But if the attacker also has the base "corecthorsebaterystaple", then they only
need to check those 600 combinations of where you put the 'r' and the 't', which is literally billions
of times easier to do than guessing "correcthorsebatterystaple" from the original list of 2000 words.
If you add this to your looping technique, then you get another factor of 23 (where you start the loop is a factor of 25, but connecting the beginning to the end reduces the previous numbers to 23 and 24 instead of 24 and 25, because now putting a letter at the beginning is the same as putting it at the end, once you tape the loop together). So now it's 23*24*25 times more difficult. That's a reasonable amount of additional entropy, but is tempered by two things:
1) Again, if the original string is lying around in plain sight somewhere, then the attacker only
needs to check those 23*24*25 combinations, which is still about a billion times easier than guessing the original string.
2) This multiplies the possible set of passwords by 13,800. One more word from your list followed by a single digit 0-9 multiplies it by 20,000 and, again, is probably far easier to remember.
I don't know what's practically best for you if you need to have a lot of information about your password physically around you in order to remember the whole thing, because I don't know how your memory works. The comic's method isn't for people who need to write down most of their password, it's for people who have trouble memorizing random strings but can easily remember short lists of words. (I've never known how "trebuchet" is mangled unless I check the comic, but I've never had any difficulty remembering that its example of a good password is "correct horse battery staple".)
Edit: I guess it was "troubador" and not "trebuchet", but in confusing those I further prove the point of the comic.