Among the thousands of observed fast radio bursts (FRBs), a few sources exhibit exceptionally high burst activity observable by many telescopes across a broad range of radio frequencies. Almost all of these highly active repeaters have been discovered by the Canadian Hydrogen Intensity Mapping Experiment/FRB project (CHIME/FRB), due to its daily observations of the entire northern sky as a transit radio telescope. FRB 20240114A is a source discovered and reported by CHIME/FRB to the community in 2024 January; given its low declination, even the detection of a few bursts hints at a high burst rate. Following the community announcement of this source as a potentially active repeater, it was extensively followed up by other observatories and has emerged as one of the most prolific FRB repeaters ever observed. This paper presents the five bursts CHIME/FRB observed from FRB 20240114A, with channelized raw voltage data saved for two bursts. We do not observe changes in the dispersion measure of the source greater than ∼1.3 pc cm⁻³ in our observations over a nearly 1 yr baseline. We find a rotation measure of ∼+320 rad m⁻². We do not find evidence for scattering at the level of <0.3 ms in the bursts, and we find no evidence for astrophysical scintillation. In our observations of FRB 20240114A, we see a burst rate ∼49 times higher than the median upper limit on the burst rate of apparent nonrepeaters also discovered by CHIME/FRB. With many thousands of bursts observed to date, FRB 20240114A has given the field an extraordinary opportunity to extensively study an FRB and understand its origins.