Water. What? This has to be one of the most unlikely concertante "instruments" - you would probably have to rank this under miscellaneous percussion. Tan Dun, best known for his film score Crouching Tiger Hidden Dragon, composed the only concerto for water and orchestra that I am aware of. It has not been released on CD, but there is a DVD of the concert as performed by David Cossin with the Royal Stockholm Philharmonic Orchestra directed by the composer. In addition, the complete concerto can be viewed in a different version on YouTube. Maybe not great music, but innovative for sure!
 Mandolin. In the baroque and classical era, this was a fairly popular concertante instrument (e.g. Vivaldi, Hummel), but as the orchestras became bigger and louder, its use almost disappeared. More recently, contemporary composers have shown an increased interest in its possibilities in concertante classical music. I have selected the concerto by Avner Dorman, performed by Eliran Avni and the Metropolis Ensemble under Andrew Cyr, available on a Naxos CD.
 Theremin.This is an instrument that was invented fairly recently (1928). The controlling section usually consists of two metal antennas which sense the position of the player's hands and control oscillators for frequency with one hand, and amplitude (volume) with the other, so it can be played without being touched. The electric signals from the theremin are amplified and sent to a loudspeaker. Like its cousin the ondes martenot, it has been used far more frequently in soundtracks than in serious classical music. I have a theremin concerto by Kalevi Aho, played by Carolina Eyck and the Lapland Chamber Orchestra under John Storgards, available on a BIS CD.
 Musical saw (or singing saw). While used occasionally by amateurs in folk music, its use in serious classical music is extremely rare. In fact, this is the first instrument in the ongoing series, where I found information about a concerto without ever having heard it - it is not in my extensive CD collection, and I cannot find it on YouTube. Some information from Wikipedia: Divination By Mirrors for Saw and Strings (1998) by Michael A. Levine is a concerto scored for two string groups tuned a quarter-step apart and placed on opposite sides of the stage with a musical saw soloist playing in both pitch universes. I would love to hear it one day.....
 Native American flute. Although this instrument has gained some foothold in New Age music, its application in classical music is of course not wide-spread. The most notable is probably its inclusion in the orchestral score of Philip Glass' second piano concerto. I know of only one concerto for it, by American composer James DeMars, titled Spirit Horses. It is played by R. Carlos Nakai and an unnamed chamber orchestra under the composer, available on a Canyon Records CD.