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Funny how Washington I-605 gets five times the discussion here over California I-605, yet only the California one was actually built.
I-605 California (link)
27.40 miles ; San Gabriel River Freeway, from I-210 near Monrovia to I-405 near Long Beach. I-605 was approved as a chargeable interstate between I-405 and I-10 on Sept. 15, 1955.  On Dec. 24, 1968, an extension was approved to I-210.  I-605 started construction in 1964, and was completed to the new 210 freeway in 1971. 
I-605 uses no control cities at all; signs say only I-605 North or South. 
When the numbering for this route was being decided (in the 1950s), California suggested "Interstate 13." 
See also: I-605 (Dan Faigin)
I-605 (proposed) Washington (link)
Proposed but not built; Seattle area. However, it may rise again. The original Route 605 was proposed as a state road, not an interstate ; however, news articles are tossing around the "I-" designation.
Route 605 (or Interstate 605) faces an uphill battle. First, it has thin support among local politicians; second, it would open up several undeveloped areas to development pressure; third, it would traverse several watersheds with protected species; fourth, it would cost at least $6 billion. 
Conceived in late '60s, cancelled in mid-'70s
A 1968 state study called for a new $151 million, 40-mile, six lane freeway east of I-405 to prevent "intolerable congestion" in South King County and the Eastside by the mid 1980s. The outer beltway was considered more palatable than another alternative: double-decking 405. Local community groups opposed 605, including a group of Bellevue homeowners from whom 1,000 showed up at a December 1968 hearing. 
In response, the state moved the proposed route east across Lake Sammamish in 1969, only to meet stiff opposition there as well. Furthermore, published reports suggested that Rep. "Concrete Al" Leland and other 605 proponents owned property near the route; the coup de grace was a later study that determined the freeway wasn't needed at all.  In 1974, regional planners facing public furor over freeways, deleted 116 miles of planned routes, including I-605. 
... back from the dead?
However, horrible I-405 traffic has stirred up talk of resurrecting 605.   Bellevue and Redmond have grown, and the old 605 route crosses WA 520 where the Microsoft campus is now. In 1998, planners were looking at WA 18, which is partially a freeway, and going north from there; or extending WA 900 along Lake Sammamish's eastern shore. 
In its 1998 highway spending bill, the state legislature appropriated $500,000 to study the issue of I-605.  
In February 2000, the draft study was unveiled, to mixed reactions. State Senator Jim Horn of Mercer Island, who secured funding for the study, said it "verifies what intuition tells you - that we need a new six-lane freeway east of 405." But others say the proposal has too many flaws: it lies outside a growth management boundary, would relieve only 3,000 cars a day from I-405, would cost $1.4 billion, and would harm endangered areas. The study did show that I-605 would remove more long-range trips than short-range, taking up to 29,000 cars a day from Interstate 5.
No specific route is laid out in the study, but a general route is shown: from Tacoma along Highway 18, near North Bend and turning north toward Monroe, then to the east of WA 203 through the Snoqualmie Valley. I-605 could then follow US 2 to Everett. The study says I-605 would carry 62,000 vehicles a day by 2020, many of them trucks; this is one-third the volume on I-5.
The Seattle Times summed up: "The next step for the I-605 study lies with the Legislature." 
In late May 2000, the I-405 Corridor Program executive committee pared its list of recommended alternatives to solve I-405 traffic problems to three: and I-605 did not make the list.  However, in late 2003, the Seattle Times stated that I-605 seemed "to have some life left, thanks to an Eastside shopping-mall developer." 
How about a $100 billion corridor?
In April 2002, the state legislative Transportation Committee revisited the 605 proposal (As George Carlin might say, they "relooked" at it.) "My vision of the world is that (I-605) would come off I-5 somewhere in south Lewis County and angle toward the foothills, and somewhere in Pierce County it would turn north and go all the way to the Canadian border," said State Sen. Swecker, part of the Transportation Committee. 
He and fellow Sen. Jim Horn envision a toll road as part of an "infrastructure corridor" carrying a freeway, rail line, pipeline and communications lines; total cost might be $100 billion. Horn suggested spacing interchanges 10 to 20 miles apart to discourage sprawl. Highway opponents aren't buying this: if 605 is built, they say, development will come. 
2004: The Washington Commerce Corridor (WCC)
By 2004, the idea had developed into something that Washington State DOT (WSDOT) says is not I-605. From its Corridor Study Fact Sheet:
The concept of the Commerce Corridor is that of a north-south, limited-access corridor that could offer additional capacity for truck and passenger traffic as well as accommodate rail and utilities. The intent is that the Corridor would begin in Lewis County and extend north to the Canadian border, and that it must be developed, financed, designed, constructed, and operated by private sector consortiums. This study is not related to the past project popularly known as "I-605;" in fact, it differs from the previous "I-605" study in many ways. The WCC freeway would be about 200 miles long, from four to ten lanes wide, and (to counter opposition) might have interchanges ten miles apart -- to start with. Though WSDOT is correct in one respect -- the WCC plan is not the same as the Seattle outer beltway plan -- the WCC's 605 ancestry is still evident, and observers seeing an advantage to conflating the two will continue to do so.
Here are some conclusions and recommendations from early studies: 
A follow-up question: given the attention to the WCC, is the Seattle-area 605 plan off the table?
About "Concrete Al"
Concrete Al Leland, like Bud Shuster, loved freeways. However, the land scandal bit him in 1970 and he lost the primary afterward. He died in 1995.
Many thanks to Josh Childers, Steve Palmer, Daniel Reece, Robert A. Norheim and Dylan Wilbanks for pointing out articles on the 605 topic.