My favorite place in LA is downtown and more specifically, Little Tokyo. There's a lot of reasons, so let me just list a perfect day for me...
Daikokuya Noodles, it's worth the wait.
Kimski Makes, best selection of baubles, vintage and new at incredibly reasonable prices.
Tokyo Japanese Outlet, great false lashes for $1-$4 and tons of dishes, notebooks and useful household knick knacks for under $10.
Grocery shopping at the Little Tokyo Galleria Market is like nothing I've ever experienced. Cheaper than the farmer's market, you almost need passport to enter. The word "organic" does not translate and I'd brace yourself for the fish tank but I can walk out with two weeks of groceries for under $60. Corner of Alameda and Third.
Grab an afternoon snack and a beer and make a few friends at Wurstkuchere. Unparalleled exotic sausages and draft options.
Kinokuniya Bookstore - magna, insane selection of notebooks and pens, endless periodical perusing
Other fun stuff;
Loft viewing - open Saturdays
Amercian Apparel outlet on 6th and Alameda, Santee Alley , FIDM scholarship store , Central Library, MOCA, The Must, Buddhist temples, Sci-Arc bookstore, Japanese Museum bookstore, Standard's rooftop bar, Urth Cafe
But I think it's just mainly that urban feel, some type of realness that only a gritty landscape brings. I love to think about the history of the buildings and be around people whose internal monologue's are so drastically different that it makes me question my own.