This is the 9710th most frequent Japanese word.
茂紀
"茂紀" is a Japanese given name and does not have a specific definition beyond its use as a name.
Here, "茂紀" seems to be a misspelling or typo used instead of a word like "証明" (documents/proof).
この活動に参加するためには、茂紀が必要です。
To participate in this activity, documents are required.
"茂紀" is used as a unique name attributed to a tree in this context.
茂紀という木が庭でとても成長しています。
The tree named "Shigeki" is growing very well in the garden.
In this sentence, "茂紀" is used as a proper noun referring to a person's name.