You want to use the union tool instead of intersect.
Then, dissolve, setting the attribute of interest as a statistics field, with MAX as the statistic type. You must leave the dissolve fields empty to get the result you want.
If you wish to retain other attributes, it could get a little tricky and could be a few extra steps. An iterated Erase might be the easiest way. If you have a lot of values and you're open to Python, it's a pretty simple script--I've already got a function to do this, and I can share it.
The difference is that intersect will retain only those areas where features overlapped; union will retain the full area, but just split features where they overlap with others.