Getty Center & Getty Villa - If you've never been to the Getty, you're missing out. Both locations are absolutely gorgeous and FREE! Parking for either location is $10 per car. For more info, visit: http://www.getty.edu/visit/
The California Science Center - Admission to the permanent exhibition galleries is free, but they do accept donations. Right now they're featuring a DEA exhibit, which is pretty fascinating. For more info, visit: http://www.californiasciencecenter.org/GenInfo/PlanningYourVisit/Hours/Hours.php
Natural History Museum - Admission is free on the first Tuesday of each month. The pictures posted in this entry are for this museum. For more info, visit: http://www.nhm.org/information/As far as parking is concerned for both the California Science Center and the Natural History Museum, you can park for free on Menlo Ave. between Exposition and King. Right in front of the Los Angeles Swim Stadium, there are several free 2 hour parking spots.
The Skirball Cultural Center - All exhibits are free on Thursdays. For more info, visit: http://www.skirball.org/index.php?option=com_content&task=view&id=26
Huntington Library - Admission is free to all visitors on the first Thursday of every month with advanced tickets. Hours on Free Day are 10:30 a.m. - 4:30 p.m. For more info, visit: http://www.huntington.org/Information/hours.html
Los Angeles County Arboretum and Botanic Gardens - Admission is free on the third Tuesday of the month (the tram is not available on free days). For more info, visit: http://www.arboretum.org/index.cfm?fuseaction=getFeature&catagory=about&FeatureID=dcc4890b-0952-11d6-a271-00d0b76949cb&PageNumber=1&CFID=1306186&CFTOKEN=35647931&StoryID=b9e905bc-1e8b-11d7-a27a-00d0b76949cb
Center Theater Group - The Ahmanson Theater, the Mark Taper Forum and the Kirk Douglas Theater make up the Center Theater Group. All 3 of them offer great shows and you can get tickets for as little as $20 through a program called Hot Tix. Apparently, the theater puts aside a small percentage of tickets for each show and sells them for $20 each. The catch is that if you order over the phone, they won't tell you where the seats are located, but if you go to the box office, they'll tell you. If you order over the phone, they'll also mail the tickets to you for free and will not charge you any lousy Ticketmaster fees. I've purchased a few of these Hot Tix seats and I've sat in the front row (on the end), in a box and in the front row, center of the balcony. Not bad, if you're on a tight budget...and aren't picky about your seats. For more info, visit: http://www.centertheatregroup.org/
I also recommend you check out http://www.goldstar.com/. They offer discount tickets to many shows, sporting events, etc in the Los Angeles area. They do have a service fee on each transaction, but its still lower than Ticketmasters' (can you tell I'm not a fan of those fees?). If you decide to sign up, please use this link: http://www.goldstar.com/tr/nc/6112/908640/?p=f908640ff26&ref=sub908640&mar=news6112
Last but not least, if you, like me, have no shame using coupons, you may want to consider getting your hands on an Entertainment '09 book http://www.entertainment.com/. The book offers some great deals (e.g. buy one get one half price or buy one get one free) for restaurants, museums, movie theaters, etc.