Final answer:
The specific area identified by Philip Hogan as the 'greatest missionary challenge of the time' is not offered in the given details, but historical context suggests regions such as Africa, where massive efforts were made to convert and modernize, were seen as critical for missionary activities.
Step-by-step explanation:
The area identified by Philip Hogan as the "greatest missionary challenge of the time" is not provided in the reference information given for this question. However, the context suggests that Christian missionaries perceived various regions as critical for their activities throughout history. Post-colonial regions in Africa, Asia, and the Pacific were significant targets for missionary work. These missionaries aimed to spread Christianity, education, medicine, and modernization. Considering this, the emphasis on the massive conversion efforts in Africa, and the continent being home to the majority of HIV-positive cases, could imply that Africa represented a significant concern for missionaries seeking to address both the spiritual and physical ailments of the time.