Of all types of beer, what type of beer require the most time to produce?
The most time? I'd think Gueze. It is a blend of 1 and 2 (and sometimes 3) year old lambic. So the minimum time to produce would be 2 years.
In addition to what's said, a major caveat is sour beer. Modern breweries have managed to coax and trick lagers into being ready without months of expensive aging, but there is no way to hurry a sour. The bacteria and wild yeast strains that make those flavors require specific conditions, and changing those conditions changes the rates of their activity and the flavor of the beer. Lambics in particular are often blended and contain "young" beer, which is only a year old, and "old" beer which can be 3 or 4 years, or more. Then the beer is aged an additional year in the bottle, but can be kept for 10 years or more.
The time it takes to make beer can be anywhere from 10 days to several months.
There are a few factors that determine how long a beer takes to be ready to drink. The original gravity of the beer (gravity measure the amount of sugar in the wort), the type and strain of yeast used, and alcohol content of the finished beer. Other factors such as any adjuncts used in the beer can affect this as well.
For example, a low gravity, low alcohol ale using an efficient yeast strain can be ready to drink in as little as 10 days. However a heavy lager beer can take 2-3 months to complete and some Belgian yeast strains can take several weeks to complete fermentation.
Some beers go through additional conditioning post-fermentation before they are at their prime to drink. This conditioning may include simple bottle conditioning or barrel-aging to impart different flavors. There are some barrel-ages stout beers that have taken 18+ months to complete the process before the brewer feels they are ready to serve.