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Ever since I saw John Cougar Mellencamp pose in front of an I-65 sign, I can't help but associate this highway with bad music.
In 1970, Alabama requested interstate funding for a 6-mile interstate link from I-65 to I-10. At the time, no number was mentioned.  In 1980, the number Interstate 210 was approved for the route. 
However, local opposition helped stop the highway just short of I-10.  As the highway would only intersect I-65 if built, its number was changed to I-165. The FHWA approved this in 1987,  and AASHTO approved the number change on June 6, 1989. 
I-165 (cancelled) Indiana
Proposed around 1980 in Indianapolis; from the I-69/I-465 interchange to the "northeast interchange," the I-65/I-70 split near downtown. That freeway was also planned as part of I-69, which even years ago was planned to extend southwest to I-70.
Word is that INDOT preferred the I-165 numbering over I-69, to avoid having to renumber all the mile markers on I-69, which start from Mile 0 at I-465.  However, if the proposed I-69 extension to Mexico takes place, Mile 0 will be relocated to the Kentucky border anyway.
From John Williams, who lives in Indianapolis:
"...In fact, the northeast interchange's T-design was built so that an eventual cloverleaf or other design interchange could be developed between I-65, I-70 and I-165. The project died and unfortunately we are left with very sharp, dangerous curves (40 mph) that have been nicknamed "Dead Man's Curve" due to the high number of fatal truck accidents shortly after the I-70 and I-65 freeways were completed downtown in 1977." 
From Michael King:
"...Interstate 69 was slated to enter Indianapolis proper and merge with I-65/70 at "Dead Man's Curve." It was to follow I-65 across the northern edge of downtown, and go south along the western side of downtown along or just west of the present location of the West Street/MLK Ave. exit, to meet I-70 southwest of the RCA Dome. I'm not sure what the alignment was to be from there, but it was originally due to leave Indianapolis and run southwesterly to Evansville along the SR 37 corridor that is presently being upgraded for the "I-69 to Mexico and Beyond" corridor." 
24.48 miles ; serves Louisville. Current traffic studies for the area include connecting Interstate 265 to nearby I-265 in Indiana. I used to consider the Kentucky and Indiana parts as the same interstate, but the FHWA lists them separately.
Kentucky's I-265 runs from US 60 near Valley Village to US 42 near Prospect. It's signed as I-265 between I-65 and I-71, and as KY 841 at the ends. Before the late 1980s, the whole route (some sections still under construction) was signed as KY 841. See also:
6.73 miles ; serves Louisville. Current traffic studies for the area include connecting Interstate 265 to nearby I-265 in Kentucky. I used to consider the Kentucky and Indiana parts as the same interstate, but the FHWA lists them separately.
The Indiana segment runs from I-64 to I-65, and was completed in 1977.  An extension to IN 62, called state route 265, opened in 1995.   The case for connecting this with Kentucky route 841 (and its I-265) is quite geometrically compelling; for transportation reasons, the two states are considering this very thing.
So how about connecting the I-265's
The Ohio River Bridges Project, shared between the Indiana DOT and Kentucky Transportation cabinet, is assessing various means of improving transportation between Louisville and southern Indiana. Among several alternatives, including mass transit and bike paths, are alignments that connect the Indiana and Kentucky I-265 segments with a new bridge over the Ohio River.
The study is a three-year effort to develop a comprehensive, detailed environmental assessment of various transportation options to connect Louisville and Southern Indiana. The project will examine alternatives such as the construction of one or more new bridges, the redesign of connecting interstate highways on both sides of the river and "non-highway" alternatives, such as mass transit, bike paths and telecommuting.
I-265 (numbered as another interstate) Tennessee
This is a former numbering for 2.25 miles of Interstate 65, from I-40 to I-24 in Nashville. It opened in downtown Nashville in 1965; the I-24/I-65 interchange had "ramps to nowhere" until the rest of the freeway opened. 
Absorbed by I-65
Tennessee rerouted Interstate 65 in late 2000, to move nonlocal traffic away from central Nashville and ease congestion. I-65 was shifted west to take over I-265's alignment, and I-265 was decommissioned. 
This renumbering was approved April 7, 2000, by AASHTO, the agency in charge of US and Interstate route numbering. 
"We recognize this in not a cure-all for congestion but by re-designating I-65 to the west, we anticipate that up to 25% of traffic using I-65 will be diverted," said Bill Moore, Tennessee DOT's Chief Engineer.
"The biggest change for local motorists will be positive. There will be less gridlock on the southeast side of the inner loop," said Transportation Commissioner Bruce Saltsman. "...When we have an opportunity to address a problem on our system by a simple change in signage, that is the route we want to go." 
57.51 miles; beltway around Indianapolis. The Indy 500 would be a little less than 10 laps. Interstate 465 was constructed between 1959 and 1970. The first section opened in 1961 and the final section -- between I-74 and I-69 -- opened in 1970.
The Dogleg, or the Spur, or...
At exit 25, the northwest corner of I-465, a 4.7-mile section of freeway proceeds due west to I-65. As of April 2002, it is called I-865. For the previous 3 decades, since its opening in 1970, it had been part of I-465, making a confusing 3-way interchange at exit 25, and creating a beltway with a tail, like an upside-down letter Q.
How this happened: I-465's original alignment included the spur, and not the circumferential section between exit 20 (I-65) and exit 25 (I-465/465/465). That was to be state route 100. In January 1970, however, 7 months before IN 100 was to open, The Indiana DOT (INDOT) convinced FHWA and AASHTO that making I-465 a complete loop, by incorporating IN 100, was a good idea. (The original request was made in 1968. ) However, the "tail" retained its I-465 designation as well, creating the 3-way confusion at exit 25.
Residents realized the spur was less inherently 465ish than the loop, and invented other names for it. INDOT listed some, maybe all, to include "I-465", "the 465 connector", "the northwest connection", "the 465 extension", "the 465 dogleg", "465 second section", [and] "the 465 ramp to 65 north."  The confusion became a public safety issue as well, as many towns know: each unclear or ambiguous place or road name can cause delays, either by having to repeat information or even backtrack from the wrong location.
INDOT did its part to help disambiguate the spur, assigning it mile markers in the 900s. Westbound traffic followed signs to "I-65 north", and eastbound to "I-465 east". ). In 1998, as part of an ITS study, it floated the idea of a better solution: a separate number. Interstate 665 was the next available even-prefixed number, but INDOT opted for 865, reasoning that "65" and "665" sounded too similar. 
This numbering change was approved by AASHTO on Oct. 11, 2002. 
R. McClelland Simpson III says that the Inner Loop works out to about 53.5 miles, while the Outer Loop is two miles longer;  divide that by 2*pi and you'd get a median about 1700 feet wide! True? Simpson explains that at certain points inner and outer 465 do separate about 100 yards apart. (US DOT mileages are centerline miles.)
They're not booing, they're chanting "Dave. Dave."
In September 2002, talk show host David Letterman started a campaign to name Interstate 465 the David Letterman Expressway.  Letterman grew up in Indianapolis and wants to give back, in a way: he offered to pay for the signs. Would I-865 become the Paul Shaffer Spur?
21.40 miles ; from I-65 near Decatur to US 72 in Huntsville. Interstate 565 was added to the Interstate system in December 1968, and was estimated to cost $60.4 million in 1970 dollars ($273 million in 2001*).
The 1980 Rand McNally map showed the proposed route running east from I-65, then looping around Huntsville to the north; but the loop part was never built. When I-565 was completed on Oct. 25, 1991, the actual cost was $478.6 million ($570M in 2001). Officially, it's the Admiral Alan B. Shepard Highway.  
The highway has four lanes, but widens to 10 lanes at its east end, which is elevated from Exit 17 eastward. 
Eastern extension planned
By 2004 or so, an extension will be constructed, a few miles eastward over Chapman Mountain, terminating at Moore's Hill Road.  However, the I-565 designation won't necessarily be extended with the new roadway; instead, it might be signed US 72. 
Western extension proposed
In summer 2003, officials in Decatur proposed extending I-565 westward along AL 20 from I-65 to US 31 in that city. The local U.S. representative has requested $30 million in federal funds for the 2.5-mile project. Reasons cited for this proposal include increasing safety and supporting new development along AL 20. 
* According to an inflation calculator at Columbia Journalism Review
4.7 miles ; a new (April 2002) designation for the spur from I-465 proper at exit 25 to I-65. . For 32 years, this spur was considered part of I-465 itself, which led to some confusion (as in I-465 leaving the exit 25 interchange in three directions). But once area motorists digest the name change, the confusion should fade away. Signs along the highway set the official change date as May 20, 2002,  though AASHTO certified the change on Oct. 11, 2002. 
For background on why I-465 ended up this way, why I-865 isn't called 665, and more, see the I-465 entry.
Interstate 865 is too new to have been included in the most recent update to FHWA's highway log, which took place Oct. 31, 2002.
Paging Patrick Fitzgerald
A small amount of controversy surrounded the April announcement of the new designation. For one, a fellow misc.transport.road "roadgeek" (search Google to find out which one) asked the Indianapolis Star if they had heard about the new number; this turned out to be a leak, as INDOT was not planning to announce it for another two weeks.  Also, some residents contended I-865 should start with an odd number since it's not a beltway or bypass; INDOT, sticking to the "letter of the law", answered that the even number is appropriate since I-865 connects to an interstate at both ends.  (For reference, the old I-465 connected to I-65 at three ends.)