"by the latest" is also correct English, but we would tend to put that phrase at the end of a sentence, and we would not commonly use "after" in either sentence. We would more likely say "Three days by the latest" or "Three days at the latest" rather than say "after two days...." However, I was wondering if they actually meant "At the latest in two days" or "In two days by the latest" which would mean within two days or before two days are over which is totally not the same as "after".
I wonder about this as well and suggest that perhaps both are different perspectives of the same idea. Each reference a point in time at the end of a span of possible points in time. To say "by the latest" is to reference the motion in the span of time and the final latest point must be inferred by the word "latest", but to say "at the latest" calls reference to the location of that final point and instead leaves the span of time to be the thing inferred... each however show the relation of the span to the final point of that span. (could one say "der letzte Moment?")
Also to allintolearning's comment I believe changing the "after" to "in" would clarify that the span is from the present moment until that final point rather than being after the point of the second day. A confusion I expressed having with the english translation elsewhere further down this thread.
The English translation given still feels a little awkward, for two reasons. First, because the "after two days" idea would usually precede the "at the latest" idea which slightly modifies it; second, because the "at the latest" idea suggests a degree of uncertainty about the time prediction which contradicts the definiteness of "after two days." I would therefore suggest. "In two days, at the latest." Examples: "Your package will arrive in two days, at the latest." "I am leaving in two days, at the latest."
Is this supposed to mean that you must do something within the next two days, or to wait for two days before doing it?
Duo suggested "At most after two days" and "At the latest after two days". I do not think I would ever use either translations because they are ambiguous, instead I'd use "at most within two days".
Unless they really mean "after two days" in which case we would say "At the latest three days", but it still doesn't make sense to me. If you are going to say "after two days" wouldn't it be no earlier than two days? So we are saying "at the latest and at the earliest at the same time? I really think "in two days" makes more sense.
it says correct solutions "in two days" and "after two days" ... those seem contradictory to me. If something comes within two days it is anywhere from the present moment to the end of the second day... if it comes after two days I wouldn't expect it before the end of that second day. To say the discussed span is ending by using the phrase "at the latest" seems confusing when combined with "after two days". I guess the unspoken part is "after two days (have passed)" is when the moment may occur "at the latest" ...
But is that then what the German sentence says? After two days have passed is then the point of latest possible occurrence?
"After two days tops" would be an accurate translation, though I'd say rather more colloquial than Duo is willing to accept.
Also, I think Duo wants you to translate "spätestens" as "at the latest," rather than, effectively, "at the most," even though here the meaning is the same.
As a native English speaker, I would say, "...two days at the latest".
Example: "The cake will be ready in two days at the latest."
"The cake will be ready at the latest after two days." feels like a botched translation or a typo. The usage of "after" goes against "at the latest" since "at the latest" or "by the latest" implies what ever happens will already have occurred at or before two days time. "After" implies that what ever event that is being talked about will occur beyond two days time at a minimum.