I am printing minis and other very detailes stuff and I find that 0.04 mm layer height gives the best looking (smoothest) result. However it takes a long time, so I am looking for ways to speed it up. Is there any slicer which offers different layer heights for the outer perimiter? So you can print your model at 0.12 mm or so, but the outer layer gets printer first 3 times at 0.04 mm?
In Ultimaker Cura, unless you print a single perimeter outline and add extra wall infill support you are not going to have different outer perimeter layer heights.
However, Cura is able to reduce the printing time, E.g. you can have fine layers for the wall (all perimeters), and coarse layers for the infill. The option is called "Infill Layer Thickness":
Infill layer thickness
Since the layer height of the infill is not important for visual quality, you can use thicker layers on the infill to reduce the print time. When adjusting this setting, always make sure that it is a multiple of the layer height, otherwise Ultimaker Cura will round it up to a multiple of the layer height. This means that you can, for example, print with an infill thickness of 0.2 mm while the layer height is 0.1 mm. The printer will first print the walls for two layers, and then it will print one thicker infill layer.
Note that this is not a standard option, you need to put Cura in the "Custom" mode and filter/search for the option using the search bar. Below you'll find the upper right corner of the Cura Graphical User Interface where I searched for the option:
Note that there are 2 options, one for regular infill of your product and one for infill of the support structures.
An other department at work is working together with a start-up university company to 3D print PEEK molds (from pellets) using 2 nozzles/extruders (to create resin injection parts), one has a large nozzle, the other has a fine nozzle. They use their own developed slicer software to use the fine nozzle for the outer contours and the coarse nozzle for infill and support. It could be that they developed this because it was not available in commercial slicers. For the 2 most common free slicers, Cura and Slic3r, there is no option to have just the outer perimeter of different height than the inner perimeters. If you think of it, it is also pretty difficult to execute, you either get:
(which does not improve the quality of the product, it will make the outside more coarse) or you'll get gaps:
Note that both upper images do not include infill, only 1 outer and 3 inner perimeters. The image below is probably what is possible with Cura, so all perimeters the same; green is infill now:
Not what you want but may still be helpful.
PrusaSlicer allows the layer height to be varied along the Z-axis so that you can have more detail in bands where it is needed, and less where it is not.
Since infill is not your time consumer, you may not find the "combine infill every n layers" helpful.