at the (very) least 表示某事物最起码的真实性、可接受程度、合意程度或肯定会发生。可以放在句首（后面用逗号），也可以放在句中（前后都用逗号）。

        at least 与句子中的其他部分之间不用逗号。该短语可以表示某事物可能或原本可能更糟、表示某人能够或应该做到的最起码的事情或者表示前一句话的真实性不确定等。

        （在《韦氏高阶英汉双解词典》中有很多例句）

at least phrase
  1. not less than a particular amount or number, and possibly more
      Mr Gray will remain the director for three years at least.
      The disease killed at least 120 people in England last year.
      We are expecting to at least double our money.
      at the very least (=not less than and probably much more than): The journey will take a year, at the very least.
      Thesaurus: too much, extra and moresynonym 
  2. even if nothing else happens or is true
      At least no one can accuse me of hiding the truth.
      You might at least have waited for me.
      We have reached an agreement that at least gives the company a chance to survive.
      at the very least: At the very least the player is guilty of violent behaviour.
      Thesaurus: true, definite and based on factssynonym 
  3. used when mentioning an advantage that exists in spite of the problem or disadvantage that you have just mentioned
      A two per cent pay rise is not very much, but at least it’s better than nothing.
      You lost, but at least you can say you tried.
      The work is difficult, but at least the pay is good.
      Thesaurus: despite somethingsynonym 
  4. used when you are saying something that changes or limits what you have just said
      The election will mean the end, or at least the beginning of the end, of the present system.
      No one saw anything, or at least they’re not talking about it if they did.
      Thesaurus: ways of adding extra informationsynonym 

后两种不适合换成at the very least。可以看出，这两种涉及某种对比。