is it "likely become" or "likely to become"?

Solution 1:

Both are correct. Become can be used as the primary verb or as an infinitive.

Examples of the use of become as a primary verb are

He becomes agitated when the music plays.

He will become agitated when the music plays.

He will likely become agitated when the music plays.

In the last two examples, become is the primary verb, and will is an auxilliary verb forming the future tense.

As an infinitive

He starts to become agitated as the music plays.

He is going to become agitated when the music plays.

He is likely to become agitated when the music plays.

In these cases, become is not the primary verb. Rather, it is used as an object of the primary verbs, starts, is going and is.

As noted in the comment by @Hellion below, both constructions have the same meaning.