Aims. HESS J1844-030 is a newly confirmed TeV source in the direction of the X-ray pulsar wind nebula (PWN) candidate G29.4+0.1 and the complex radio source G29.37+0.1, which is likely formed by the superposition of a background radio galaxy and a Galactic supernova remnant (SNR). Many scenarios have been proposed to explain the origin of HESS J1844-030, based on several sources that are capable of producing very high energy radiation. We investigate the possible connection between the SNR, the PWN G29.4+0.1, and HESS J1844-030 to shed light on the astrophysical origin of the TeV emission. Methods. We performed an imaging and spectral study of the X-ray emission from the PWN G29.4+0.1 using archival observations obtained with the Chandra and XMM-Newton telescopes. Public radio continuum and HI data were used to derive distance constraints for the SNR that is linked to G29.37+0.1 and to investigate the interstellar medium where it is expanding. We applied a simple model of the evolution of a PWN inside an SNR to analyze the association between G29.4+0.1 and the radio emission from G29.37+0.1. We compared the spectral properties of the system with the population of TeV PWNe to investigate if HESS J1844-030 is the very high energy counterpart of the X-ray PWN G29.4+0.1. Results. Based on the morphology and spectral behavior in the X-ray band, we conclude that G29.4+0.1 is a PWN and that a point source embedded on it is the powering pulsar. The HI data revealed that the SNR linked to G29.37+0.1 is a Galactic source at 6.5 kpc and expanding in a nonuniform medium. From the analysis of the pulsar motion and the pressure balance at the boundary of X-ray emission, we conclude that G29.4+0.1 could be a PWN that is located inside its host remnant, forming a new composite SNR. Based on the magnetic field of the PWN obtained from the X-ray luminosity, we found that the population of electrons producing synchrotron radiation in the keV band can also produce IC photons in the TeV band. This suggests that HESS J1844-030 could be the very high energy counterpart of G29.4+0.1.