まい

Not Likely, Don’t intend to
Dictionary form verb + まい
一段 Verbs – ます stem + まい or Dictionary + まい
する – すまい
くる – こまい

まい is the negative volitional form. It is old Japanese, and is pretty rare. You are most likely to encounter it in literature or period pieces.
It mostly means what you would expect:

• Probably not
• Don’t intend to
• Don’t (With a command, rare, probably not on JLPT)
• ではあるまいし – Not like its a….
• くれまいか – Won’t you
• ようがまいが – Do or don’t, same as にしろ

Although 一段 verbs are supposed to use the ます stem + まい, they are very frequently just dictionary form + まい. (Source 1, Source 2, Source 3)

Probably not

ないでしょう
Dictionary form verb + まい
Noun +ではあるまい

As you should expect from the negative volitional form, まい means ”~ないでしょう” (Not likely).
This is the opposite of words like しましょう。

Nouns use ではあるまい (From ではない).
ではあるまい may also appear as じゃあるまい.

Examples

• 二度とするまい – That probably won’t happen again.  
  (二度としないでしょう)
• あれはりすではあるまい – That’s probably not a squirrel.
  (りすではないでしょう)
• 彼は殺人鬼ではあるまい – He’s probably not a murderer. (Sounds more certain than the English).
  (殺人鬼ではないでしょう) 
• 城はもう守るまい – They probably won’t protect the castle anymore
  (もう守らないでしょう)

Don’t intend

ないつもりだ
Dictionary form verb + まい

This usage means “Don’t intend to”. Its almost identical to the “Not likely” meaning.
The “don’t intend” usage doesn’t really use ではるまい and tends to be used with words expressing a decision, or “not anymore“; and/or an emphasis particle like よ.
But you’ll have to use context to determine whether its “Don’t intend” or “Not likely”.

Examples

• もう行くまいと決めた – I decided to not go again. 
  (行かないつもりだ)
• 何を言われても彼女を疑うまい. – No matter what is said, I don’t intend to doubt her.
  (疑わないつもりだ)
• もう自分の夢を諦めまい – I don’t intend to give up on my dreams anymore. 
  (諦めないつもりだ)

Don’t

てはいけない
Dictionary form verb + まい + command

Probably not on the JLPT
This meaning is much rarer than the others and I doubt you’ll encounter it unless you are reading a novel set in the past. This pattern explicitly needs to be used with a command.
As this is old Japanese, the most common command word paired with it is “ぞ”.

Examples

• 笑うまいぞ (Weblio) – Don’t laugh
• 彼の前は絶対に言うまいぞ – Definitely don’t say that in front of him. 
• 庭の果物を取るまいぞ – Don’t take the fruit in the garden

ではあるまいし

Its not like (it’s) a…
Noun + ではあるまいし

This is N1 grammar, but it seemed silly to make it a separate page.
This pattern is essentially ではるまい (Not likely) + し (things left unsaid).
It is used to express “Its not like you’re a child, etc.
This pattern usually expresses criticism, and is almost exclusively spoken. 
Tends to be used with a command. 

This may appear as じゃあるまいし, or じゃないし.

Examples

• 子供ではあるまいしきちんと野菜を食べなさい – Its not like you’re a child so eat your vegetables.
• 死ぬわけではあるまいし、早くやりなさい – Its not like you’re going to die, hurry up and do it.
• 飛行機ではあるまいし、運転できるはずだ – Its not like its a plane, you should be able to drive it.
• 神ではあるまいし、そんなことは約束できないよ – Its not like you’re a god so you can’t promise that.
• 恥ずかしいことではあるまいし、先生に聞いたら？ – Its not like its something embarrassing, why don’t you ask the teacher?

くれまいか

Won’t you/ませんか

くれまいか is the same as くれないでしょうか/ませんか

Examples

• 譲ってくれまいか – Won’t you give it to me
• てつだってくれまいか – Won’t you give me a hand.
• 一緒に行ってくれまいか – Won’t you go with me?
  
  

ようがまいが

Do or Don’t, doesn’t matter

This is just a variation of the にしろ pattern.
Its slightly less common than にしろ.
If I had to describe a nuance difference, its that にしろ frames it as choosing, while ようがまいが frames is more indifferent.
You can replace が with と.
This pattern may also appear as NounだろうがNounだろうが.

Examples

• 来ようがこまいが、関係ない – Come or don’t, it makes no difference.
• 猫だろうが虎だろうが, ただの猫だろう – Cat or tiger, Its still just a cat.
• 食べようと食べまいと、注文した分を払わないといけない – Eat it or don’t, you have to pay for what you ordered. 
• 入院されようがされまいが、連絡すべきだった – Hospitalized or not, you should have contacted us.