TLDR; it depends on what you want to grow, but generally speaking a depth of 20-25 cm (8-10 inches) suffices to grow many vegetables and herbes, but do investigate if your roof supports such a weight because a rooftop garden can become very heavy when it's soaked with water!
Once you are set up, rooftop gardening is similar to container gardening. The main difference is that there usually is more wind and less shade compared to a ground-level garden. However, before you can start there are quite a bit of things to consider:
Check if you have proper access to the rooftop as you'll want to plant, water, add fertilizer, remove weeds and harvest regularly. You don't want to do this balancing yourself on a ladder for example. Also consider how you are going to get all materials (most notably water!) on your rooftop.
Check if local ordinances and home owner association rules allow for the construction of a rooftop garden and if so under what conditions.
Check if the roof and building infrastructure supports the weight of the garden depth you have in mind. The larger and deeper your garden, the heavier it becomes especially when it is soaked with water after rainfall. You might want to hire a professional to check this for you.
Make sure you have a proper drainage system for draining superfluous water from rainfall and check if it isn't clogged up every once in a while.
Figure out the needs for each plant you want to grow. Make a list of everything you want to grow and investigate the required minimum soil depth (just google it), but also other relevant requirements, e.g. how much water and sunlight does the plant need for proper growth and can it stand windy conditions that are prevalent on rooftops?
You mentioned a few plants you want to grow:
Chives can grow in as little as 10 cm (4 inch) of soil and are very easy to keep.
Strawberries require at least 15 cm (6 inch) of soil and will make excellent rooftop plants if you can keep them out of the wind. Think about whether you want june-bearing (harvest once a year), everbearing (harvest twice a year) or day-neutral (harvest all summer) strawberries.
Carrots are fairly easy to grow and require around 20 cm (8 inches) soil depth.
Potatoes can be grown using shallow planting or deep planting. For shallow planting you plant about 10-15 cm (4-6 inch) deep and as the plant grows you add soil on top. Basically you create a tiny hill around the plant so the potatoes themselves remain covered. This is necessary because when potatoes are exposed to sunlight they create toxins and become inedible. Deep planting requires less maintenance but you'll want a minimum of 20-25 cm (8-10 inches) soil depth
You also mentioned you don't like re-planting, but crop rotation is a must if you want to grow potatoes (a second time). More info about growing potatoes in a container.