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"You know I take the 110 to the 105, get off on Crenshaw tell my homies look alive" - Skee-Lo, "I Wish"
I-105 is featured prominently in the actioner (inadvertent) parody Speed. In the movie, the bus Evel Knievels a 50-foot gap in an otherwise finished connecting ramp (a Caltrans oversight?). In real life, the bus would fall about four feet in that interval, striking the bridge structure head-on and killing all aboard.
17.32 miles ; from I-605 in Norwalk to LAX. Formally the Glenn Anderson Freeway/Transitway but commonly called the Century Freeway (or simply The 105), I-105 is the last interstate planned for the LA area. (Unless you count the Interstate 210 extension.)
The freeway had been on the drawing board since 1958, and originally was to extend farther east. I-105 was added to the Interstate highway system in 1968.  By 1970, the proposed route was as it is now, and the cost was estimated at $190 million. 
In 1972, environmentalists and residents of the Century Freeway corridor sued to block the freeway. In 1979, a Consent Decree was issued altering the size and scope of the freeway, and creating housing and employment programs. The first construction project broke ground on May 1, 1982. In 1987, I-105 was renamed in honor of Congressman Glenn M. Anderson. 
I-105 opened at 3:13 pm on October 14, 1993, at a cost of $2.3 billion. There are six traffic lanes, and two HOV lanes with their own exit ramps. There are 5-level interchanges at I-405 and I-110, the fifth level arising from the direct access between commuter lanes on the intersecting freeways. The LA Metro commuter line runs in the median.
Public Roads notes that I-105 has been called Los Angeles County's "first high-tech roadway"; pavement sensors, video cameras, and a linked computer system helps technicians monitor traffic flow and respond to incidents.
About Congressman Glenn Anderson
The highway's namesake, representing the South Bay-Mid Cities portion of Los Angeles County, was instrumental in getting federal funding for various transportation projects in the region, including the Metro Red Line subway and the 105 freeway. He served as Chairman of the House Committee on Public Works and Transportation, a post later held by the infamous Bud Shuster. He also helped get CA 17 upgraded to Interstate 880 in the 1980s. He retired in 1993 and died in 1994. 
Audit faults CalTrans as roadway sinks
Portions of the I-105 shoulder between the Los Angeles and San Gabriel rivers are starting to buckle and sink because of high groundwater, the Los Angles Times reported in March 1999. A report issued by the Bureau of State Audits in August of that year attributed the problem to a decision in the 1960s to build part of I-105 below ground level to mollify local groups opposing the freeway.
CalTrans has been pumping water away from the area (in secret, until the Times made it public); the pumping, along with repairs to the roadway, are anticipated to cost almost $120 million. According to the audit report, CalTrans officials omitted soil and water tests, recommended by staff members, that would have predicted the problem. 
The 105/110 interchange
This "stack" interchange, normally four levels, has a fifth level in a direct HOV-to-HOV connecting ramp, reaching 122 feet above ground level. The interchange won an Award of Merit in Category 1 - Urban Highways from the FHWA in 1996.
Gabe London writes in "Zen and the Art of the Freeway Interchange:
"Ah, freeway bliss. If Martians landed here I would be proud, because this man made wonder is functional art on a epic scale... The 110's ramps shoot up and split East and West like a concrete goddess emerging from the Hades of our meager landed existence. At 90 mph and nearly nine stories in the air YOU ARE FLYING, and that's freeway Zen.
"Even if you're alone, I recommend you risk the ticket in the carpool lane to reach for the smoggy sky on your way up, up, up those g-dd-mned beautiful ramps."  [name in vain edited for airline use by Kurumi]
I-105 (decommissioned) California
In the mid-1960s, the I-105 number was used for the beginning of the Santa Ana Freeway from the Hollywood Freeway to I-10 -- the left side of the thin triangle US 101, I-5, and I-10 form. However, it was never signed. In 1968, that section reverted to US 101. 
3.49 miles ; runs northwest from the I-5/OR 126 interchange in Eugene. The city of Eugene reports I-105 being completed in 1961, the same time as I-5 in the area.  But the state DOT says I-105 was completed in 1969.  Or 1970.