Housing unweaned calves individually in barren environments negatively affects their growth, cognitive ability, and adaptability to environmental changes in later life. Social housing has been shown to improve those aspects, whereas physical environmental enrichment has rarely been studied in calves. Little is known about whether the combination of both components offers further benefits. Furthermore, curiosity has been considered an intrinsic factor underpinning cognitive performance, which has yet to be determined in calves. The first objective was to compare the effects of providing physical enrichment items and pair housing calves before weaning, and their combination on the weight gain, behaviour and cognitive ability of calves once regrouped after weaning. The second objective was to investigate if calves’ exploratory behaviour in a novel environment can predict their cognitive ability. Forty-eight Holstein calves were allocated to eight groups 2 days after birth. Within each group, two calves were assigned to individual pens and four to two pair pens. One individual pen and one pair pen within each group were provided with brushes, chains, teats, and nets filled with strawberry-scented hay as physical enrichment items. Remaining pens received no additional enrichment items. All calves from a group were introduced to one post-weaning pen when the youngest calf was 9 weeks of age. Calves were weighed on days 1 and 7 in post-weaning pens. They were video-recorded on days 1, 3 and 11 and behavioural data were collected. Spontaneous object recognition tests were conducted within one week after behavioural data collection to assess cognitive ability in terms of how long after exposure calves recognised objects, indicated by differential expression of exploratory behaviour. Physical enrichment items and pair housing had interactions on average daily gain (F1,33 = 5.460, p = 0.026), with calves in physically enriched pair pens showing higher average daily gain than those in non-enriched pair pens and tending to show higher average daily gains than those in physically enriched individual pens. Physically enriched calves expressed more exploratory behaviour and social sniffing than non-enriched calves (F1,111 = 20.691, p < 0.001; F1,111 = 14.433, p < 0.001). Pair housed calves spent more time cross-sucking than individually housed calves (F1,111 = 8.848, p = 0.008). Compared with non-enriched calves, physically enriched calves were more inclined to explore the novel object than the object already presented 15-min ago (χ2 = 3.282, df = 1, p = 0.070). There was no association between exploratory behaviour upon initial introduction to post-weaning pens and performance in object recognition tests. In conclusion, the combination of physical enrichment and pair housing improves calves’ average daily gain after weaning when compared with either component alone. Physical enrichment seemed to improve calves’ memory and adaptability to change, whilst pair housing did not. Calves’ exploratory behaviour in novel environments may not contribute to their cognitive performance.