You're The Guy I Want To Share My Money With (with William S. Burroughs and John Giorno, 1981). Contains rough versions of tracks from Laurie Anderson's United States Live show plus a number of tracks of Burroughs telling his wry stories and two unlistenable John Giorno long-form poems. They might be good, but I can't tell because both use voice modulators that have me reaching for the forward button faster than the opening chords of "Hotel California."
Big Science (1982). Anderson's proper debut album is disorienting, psychedelic, witty, chilly, wise, even profound. It should be clear that I love it wholeheartedly. Although it consists entirely of reworked parts of her United States show, it never sounds like the distillation of anything. Except in the sense that it leaves me intoxicated. The first track is "From The Air," in which Anderson uses her authoritative and calm voice to guide listeners through the weirdest crash landing procedure, culminating in the lines: "put your hands over your eyes/jump out of the plane/there is no pilot/you are not alone." That reversal between crisis and comfort gets right at the heart of Anderson's brilliance: everyday is a crisis, a constant teetering over an unfathomable abyss that all authorities spend their entire careers trying to deny, and yet there is some strength to be reached in the notion that others are in this with you. And then, of course, there's "O Superman," her ode to misuse of authority, so scalding that I'm surprised that Ronald Reagan was ever able to return to his office after this song came out. Seems like the smart thing to do would have been to retire with some remaining semblance of dignity.
Mister Heartbreak (1984). Her second album borrowed more parts of United States Live and added Bill Laswell (last seen here in yesterday's Last Exit video) as co-producer and bassist, Adrian Belew on guitar, and a bunch of other people, including William Burroughs and Peter Gabriel on vocals. The songs include "Gravity's Angel," which mentions and seems inspired by Pynchon's Gravity's Rainbow, although it doesn't borrow imagery quite as directly as Wikipedia suggests, and "Langue d'Amour," a love song like no other. But it's not quite as great as Big Science.
Talk Normal: The Laurie Anderson Anthology (2000). I'm not sure how well this one works overall. I snatched up the tracks from United States when it was unavailable some years back. Now I don't need them, but I haven't erased them because, well, I paid for them, y'know.
And simply for your pleasure, here's a couple of videos Anderson made of her and her, well, clone: