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The phrase "on the up-and-up" means "legitimate" or "legal" or "reputable" or, to use another idiom, "above-board". For instance:

Although Pete didn't look like a city official, Joe assumed his offer to sell the Brooklyn Bridge was totally on the up-and-up.

Where does this expression come from?

Related: What does “We don’t do anything that’s not completely up and up” mean?

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Here's some stuff about the usage you refer to - "whose exact origins are mysterious", apparently. I've never heard the expression used thus. So far as I'm concerned, it's always meant "increasing continuously/exponentially". I think this "honest" sense (to me, "on the level", not going upwards at all) is weird/contrary. To be honest, I don't care where it came from - I just don't want to use it, because it conflicts with a meaning I already know. – FumbleFingers Aug 14 '12 at 1:04
up vote 4 down vote accepted

Apparently its origin isn't clear. Grammarist.com reports:

The idiom on the up and up­­—whose exact origins are mysterious, though it dates from the late 19th century, is likely American, and appears to come from sports betting—means (1) open and honest, legitimate; and (2) on the rise. Though some dictionaries promote one or the other definition (usually the first), both senses are so widespread that we have to accept them.

The article then provides three citations for the first meaning, dated 1929, 1937, 1938, and one 1949 citation for the latter meaning. Merriam-webster.com says “First Known Use of UP-AND-UP 1863” for the meaning “an honest or respectable course”, but gives no citations. Languagehat.com has numerous comments on the two different meanings, but nothing about etymology; and Geoffrey Nunberg uses “on the up and up” as an example of a phrase that many people use one way, and many another, with people in either group not knowing of the other.

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I just found this 1929 Washington Post "slang dictionary" in a local history blog which claims that "on the up and up"(1) was slang unknown to most adults at the time. – Charles Aug 29 '12 at 21:48

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