What's the Name of That Book??? discussion
SOLVED: Children's/YA
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SOLVED: Boy runs away, lives with dumpster divers [s]
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Searching through snippet views on Google Books, this seems to match with your general description. It talks about a boy spending a summer with Mr. and Mrs. Katz (who do appear to be dumpster divers). It talks about a garage that can't be used because it's too filled with stuff. There are references to using mnemonic devices to remember a grocery list "One is bun...two is shoe..."
There is even a reference to a "mile-high banana cake". Quote: "Instead of black spotted bananas he fancied he saw a mile-high banana cake."

I've added it to the bookshelf -- now this thread just needs to be updated and moved to the Solved folder. :)

He meets an odd man by the dumpster, who is looking for stuff - they never call it dumpster diving in the book.
He decides to stay with the man and his wife, and his room is filled with odds and ends. I think the "One, Two, Buckle My Shoe" rhyme is hung behind the door.
The couple scrounges for everything; all their house repairs are done with old junk. One or both of them uses mnemonic devices to remember things.
One big scene I remember is when Henry and the man tackle clearing out the shed, which hasn't been opened in years because it's so full of stuff.
The other is when Henry's dad finds him, and they've all just made a huge banana cake, and he flips out.
This was a strange but sweet book; Henry never thought the dumpster divers were bad or gross, just odd.