  We report the results of three Chandra observations covering most of the
extent of the TeV gamma-ray source HESS J1616-508 and a search for a lower
energy counterpart to this source. We detect 56 X-ray sources, of which 37 have
counterparts at lower frequencies, including a young massive star cluster, but
none of them appears to be a particularly promising counterpart to the TeV
source. The brightest X-ray source, CXOU J161423.4-505738 with a flux of
5x10^{-13} c.g.s. in the 0.5-7.0 keV band, has a hard spectrum that is well fit
by a power-law model with a photon index of 0.2+/-0.3 and is a likely
intermediate polar CV candidate. No counterparts of this source were detected
at other wavelengths. CVs are not known to produce extended TeV emission, and
the source is also largely offset (19') from HESS J1616-508, making them
unlikely to be associated. We have also set an upper limit on the X-ray flux of
PSR J1614-5048 in the 0.5-8 keV band (F<5x10^{-15} c.g.s. at a 90% confidence
level). This makes PSR J1614-5048 one of the least X-ray efficient pulsars
known, with an X-ray efficiency {\eta}=L_x/Edot < 2x10^{-5} in the 0.5-8 keV
band. We find no evidence supporting the association between the pulsar and the
TeV source. We rule out a number of X-ray sources as possible counterparts to
the TeV emission and do not find a plausible counterpart among the other
sources. Lastly, we discuss the possible relation of PSR J1617-5055 to HESS
J1616-508 in light of the new observations.
