The join hint restricts the number of possible plans which the optimiser has to evaluate to the point where it has time to pick a suitable plan, hence with the hint you're no longer getting a timeout. It's still a horrible plan - you're reading the same tables again and again. If you post up the view definitions, it might be possible to work through them to generate a cleaner query with less reads of the same tables.
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Understanding and using APPLY, (I) and (II) Paul White
Hidden RBAR: Triangular Joins / The "Numbers" or "Tally" Table: What it is and how it replaces a loop Jeff Moden