Juse seen this thread; I wanted to make a couple of comments...
First in Bristol have single and return tickets on routes they split in the city centre a few years ago that allow travel on one route as far as the Tramways Centre and then back out on the other side.
Do the Night Buses in Bristol not count as a route where "break of journey" is not only permitted, but even
encouraged? You have a flat fare which allows travel from any one point in the city to another, breaking the journey in the centre where required. Obviously it's not perfect - it's not as though you can jump off at Brislington and jump on the next an hour later - but it's a start! I think it would be useful if operators would permit it properly though, even if it caused some fares to go up. It's especially true in areas where the local hospital(s) have their own dedicated routes as is true here - you could board my Uni bus, the SPA1, in Oldfield Park and then change at Lower Weston to the 14 to the Hospital. Having said that though, Bath has a circular route which pretty much does this anyway (the 20A/C)...
I'm not sure of this officially but I believe there must have been this option in Southend. The 7/8 routes operated by Arriva follow outwardly similar routes except the 7 serving Ashingdon and the 8 serving Hawkwell, however the fare stages on the ticket machines for the number 7 also included those fare stages on the number 8. Not sure if this was ever used in practice but assume it must've been possible somehow.
Knowing AS, I think not

Knowing Southend Transport before them though, it is feasible. But gone are the days of 15p Child singles in Southend (late 90s); the same journey would now cost about £1. On the other hand, I'm not sure Southend would be a good place to try the BOJ rule. There are few places it would work. It would be a lot in Chelmsford I think, especially on the 42.
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O/T: I am still struggling to believe that Southend actually has 12 brand new Versas, even though I been on a couple of them! Not too surprisingly though, I've already spotted one on a route besides the 9 (the 29) - and have travelled on a Dart on a 9.