The words in the above headline all mean "at least," but it can be confusing as to which context is correct for the right word. Note too that "at least" is a prepositional phrase in English, whereas in German the expression is usually a simple adverb. The adverb mindestens is probably the most common:
In Deutschland ist es so: Asylbewerber müssen mindestens drei Monate warten.
In Germany, it's like this: Asylum applicants must wait at least three months.Play Caption
The adverb mindestens usually, as in the first case above, refers to a length of time or an amount of something:
Heutiger Hochwasserstand: wieder mindestens zehn Zentimeter.
Today's flood water level: at least ten centimeters again.
Caption 44, Die Klasse - Berlin '61 - Part 2Play Caption
It's used somewhat less often to refer to circumstances:
Zwar haben die Zuwanderer in der Regel eine hohe oder auch mittlere Qualifikation, die also mindestens einem deutschen, äh, Abschluss, [Bundesagentur für Arbeit Hessen].
Indeed, the immigrants normally have a high or also mid-level qualification that is at least on par with a German, uh, degree, [Federal Agency for Employment, Hesse].
Captions 35-36, Rhein-Main-TV aktuell - Mehr Beschäftigung in Rhein-MainPlay Caption
The related adverb zumindest, on the other hand, is used much more commonly for situations rather than lengths of time or amounts:
Nun, zumindest habe ich meinen ersten Anruf hinter mir.
So, at least I have my first call behind me.
Caption 69, Berufsleben - das Vorstellungsgespräch - Part 4Play Caption
Be careful not to jumble mindestens and zumindest into one (incorrect) word, something that even native German speakers occasionally do!
Eislaufen ist leicht, zumindestens [sic, zumindest] leichter als auf Vanessas Party eingeladen zu werden.
Ice skating is easy, at least easier than getting invited to Vanessa's party.
Captions 82-83, Küss mich, Frosch - Leb wohl, kleiner Prinz - Part 3Play Caption
So remember that mindestens and zumindest are real words, either of which would have been correct in the above sentence, but "zumindestens" is not a proper word at all!
Even less common, but making things even more complex, is the adjective mindeste / mindester / mindestes. This adjective can also be nominalized, or turned into a noun, such as das Mindeste (the least thing), or zum Mindesten (at least).
Go to Yabla German and find examples of the above adverbs and adjectives to see how they are used in a real-world conversational context. To go even deeper into the adjectival usage, read the Duden page for mindeste / mindester / mindestes.