The Li+ - (H2) n n=1-3 complexes are investigated through infrared spectra recorded in the H-H stretch region (3980-4120 cm-1) and through ab initio calculations at the MP2aug-cc-pVQZ level. The rotationally resolved H-H stretch band of Li+ - H2 is centered at 4053.4 cm-1 [a -108 cm-1 shift from the Q1 (0) transition of H2]. The spectrum exhibits rotational substructure consistent with the complex possessing a T-shaped equilibrium geometry, with the Li+ ion attached to a slightly perturbed H2 molecule. Around 100 rovibrational transitions belonging to parallel Ka =0-0, 1-1, 2-2, and 3-3 subbands are observed. The Ka =0-0 and 1-1 transitions are fitted by a Watson A -reduced Hamiltonian yielding effective molecular parameters. The vibrationally averaged intermolecular separation in the ground vibrational state is estimated as 2.056 Å increasing by 0.004 Å when the H2 subunit is vibrationally excited. The spectroscopic data are compared to results from rovibrational calculations using recent three dimensional Li+ - H2 potential energy surfaces [Martinazzo, J. Chem. Phys. 119, 11241 (2003); Kraemer and Špirko, Chem. Phys. 330, 190 (2006)]. The H-H stretch band of Li+ - (H2) 2, which is centered at 4055.5 cm-1 also exhibits resolved rovibrational structure. The spectroscopic data along with ab initio calculations support a H2 - Li+ - H2 geometry, in which the two H2 molecules are disposed on opposite sides of the central Li+ ion. The two equivalent Li+ H2 bonds have approximately the same length as the intermolecular bond in Li+ - H2. The Li+ - (H2) 3 cluster is predicted to possess a trigonal structure in which a central Li+ ion is surrounded by three equivalent H2 molecules. Its infrared spectrum features a broad unresolved band centered at 4060 cm-1.