... かれ、その老夫の答へ申ししく、「あは、国つ神大山津見の神の子ぞ」。わが名は足名椎と言ひ、妻が名は手名椎と言ひ、娘が名は櫛名田比売と言ふ。 ―「古事記」の「八俣の大蛇」より
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This particular しく is not the 連用形【れんようけい】 of adjective ending し (modern しい). Instead, this is past recollective auxiliary き in the 連体形【れんたいけい】 + the ク語法【ごほう】, a.k.a. "-ku nominalization" pattern similar to uses of modern こと, seen in Old Japanese and persisting in certain set expressions even into modern usage, such as 曰【いわ】く ("history; reasons, background", literally something like "that which is said [about something]") or 恐【おそ】らく (originally "that which is frightening", used in modern Japanese to mean "probably"; compare English "I'm afraid it's the case that...").
⇒ So 申【もう】ししく is ultimately equivalent to modern 申【もう】したこと.
Let's look more closely at your problem word, 申【もう】ししく. This breaks down as follows:
[VOWEL]+[VOWEL]diphthong was not allowed in Old Japanese phonology, so either one vowel eclipses the other, or the two fuse.