In this paper, we investigate the performance of hybrid automatic repeat request (HARQ) with incremental redundancy (IR) and with code combining (CC) from an information-theoretic perspective over a point-to-point free-space optical (FSO) system. First, we introduce new closed-form expressions for the probability density function, the cumulative distribution function, the moment generating function, and the moments of an FSO link modeled by the Gamma fading channel subject to pointing errors and using intensity modulation with direct detection technique at the receiver. Based on these formulas, we derive exact results for the average bit error rate and the capacity in terms of Meijer's G functions. Moreover, we present asymptotic expressions by utilizing the Meijer's G function expansion and using the moments method, too, for the ergodic capacity approximations. Then, we provide novel analytical expressions for the outage probability, the average number of transmissions, and the average transmission rate for HARQ with IR, assuming a maximum number of rounds for the HARQ protocol. Besides, we offer asymptotic expressions for these results in terms of simple elementary functions. Additionally, we compare the performance of HARQ with IR and HARQ with CC. Our analysis demonstrates that HARQ with IR outperforms HARQ with CC.