Not been to LAX for long time - (2003) - there was...
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(1) The Proud Bird, near threshold of runways 25L/R - (Restaurant + Bar)
(used to be owned by retired Continental pilot)
(2) The bar in Hacienda Hotel, on S. Sepulveda - 1/2 mi from airport
(their restaurant La Mariposa is open all night, inexpensive)
(3) 1 block South of Hacienda, same side, small bar owned by Chinese lady
(cannot recall name - inexpensive food and beer - the only bar there)
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(3) Does the "Grand Bar and Grill" ring any bells ?