When to use "most" or "the most"

"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

'Most' is the superlative form of "much and many" and used as adjective, adverb and noun.

As noun:

The most I can offer for the house is $100,000.

(Don't put forth the argument adjective & article combination to be noun.)

The superlative with "most" is sometimes used when there is no ideas of comparison, but merely a desire to indicate a quality in a very high degree. (Grammar Ref.)

This is most convenient.
This is a most wonderful sight.
It was a most eloquent speech.

It should be noted that even though 'most' is in superlative form, 'the' is not used before it. Sometimes, 'most' goes without article or sometimes it takes 'a' instead of 'the'.

In your example it is an adverb; and without 'the' amply conveys what is meant.----there is no idea of comparison.