Which of the following brewing methods is least likely to produce a sediment-free cup?

The French press brewing method is least likely to produce a sediment-free cup because it involves steeping coffee grounds directly in hot water and then pressing them to the bottom with a metal or plastic mesh filter. This method does not completely separate the coffee from the grounds, and as a result, fine coffee particles can easily pass through the filter, leading to sediment in the final brew.

In contrast, pour-over methods utilize a paper or specialized metal filter that typically traps more fine particles, resulting in a cleaner cup. The Aeropress also employs a paper filter, which creates a barrier that significantly reduces sediment. Cold brew methods, while potentially allowing some fine particles through, often steep the coffee for extended periods and are typically filtered multiple times, further reducing sediment. Therefore, among these options, the French press is the most prone to producing sediment in the finished cup.