"But what I remembered most is moving a lot" is correct, with or without "the". Although "the most" is the superlative, preferable.

Here, "most" is used as an adverb modifying the verb "remember", meaning "to the greatest extent". There may be other examples, where it can mean "extremely" as in the following:"it was most kind of you", "that is most probably correct".

As an adjective while qualifying a noun, it takes "the" as in the following example: Here's the most expensive book I ever bought.

As a noun, "most" takes the definite article. For example, "The most (that) you can do is to try again."

For further review: http://www.oxforddictionaries.com/definition/english/most http://dictionary.cambridge.org/grammar/british-grammar/most-the-most-mostly