How About Half a Kilometer
The GRACE mission was mentioned in a very complete answer by @Rob (mentioned as GRACE-FO, which is the current mission replacing GRACE). However, unmentioned is the fact that the satellites swapped positions periodically. Since the satellites use K-band (and laser for GFO) ranging, the rear satellite was exposing its K-band "horn" to bits of atmosphere at the low mission altitudes (in the 300-400 km range). This was expected to be a possible cause of instrument failure and thus the satellites (referred to as GRC-A and GRC-B) periodically swapped positions in orbit.
During this maneuver, the trailing satellite "overtakes" the leading satellite and both rotate to keep the K-band instruments facing each other. This is a quality summary of the maneuver done in December 2005 (poor web formatting but oh well), and it is noted that the satellites had a range of 406 meters at closest approach.
This link suggests that during a 2015 swap the range at closest approach was under 300 meters, but there's really not much more detailed information.
GRAIL was a GRACE-like mission in lunar orbit, which flew at a similar separation. Since there is no lunar atmosphere, the spacecraft did not swap positions to my knowledge. However, this did allow them to fly extremely low, producing excellent maps of the lunar gravity field.
GRACE-FO is currently flying, and was doing on-orbit checks during its early mission phase. It is quite possible that the two satellites came even closer during checks, but I can't find any reliable information on that.