ますます

ますます means "more and more", "increasingly": something that is already the case grows further in the same direction. It is an adverb, so it goes before the verb or adjective, which is almost always a word of change – なる, ふえる増える, きごう〜てくる, きごう〜ている.

The direction can just as well be downward. English then says "less and less", "fewer and fewer" or "worse and worse".

In writing, ますます is sometimes spelled ますます益々.

ますます or だんだん

だんだん describes a change that comes step by step, and it can start from nothing. ますます starts from a state that already holds and says there is more of it now.

In formal letters

A formal letter often opens by wishing the reader continued success. There ますます comes before a noun of state rather than a verb of change, and the whole phrase is a set formula.