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Both sentences could mean the same thing, however I like you best. I like chocolate best, better than anything else can be used when what one is choosing from is not specified I like you the best. Between chocolate, vanilla, and strawberry, I like vanilla the best can be used when choosing from some choices.
Understanding the Context
Your example already shows how to use "best" as an adverb. It is also a superlative, like "greatest", or "highest", so just as you would use it as an adjective to show that something is the ultimate example of it's kind when used as an adverb you do so to indicate that the adjective it precedes is to the highest degree possible. In your example "experienced" is the past tense of the verb to ...