My own Ubuntu started with Ubuntu 17.10 (pre-release) where I added LXDE (Lubuntu), XFCE (Xubuntu), MATE (Ubuntu-MATE). It's now been release-upgraded a few times and am now on 20.04 with the LXDE being replaced by LXQt.
As I don't have, nor have ever added Budgie/Trinity to anything - I can't speak to those.
In my experience when you add the second desktop you'll be unlikely to have problems, the chance of problems increases as you add more.
I had problems with KDE & MATE on my box (LXDE & XFCE & GNOME already there), but if [all] my other desktops weren't installed KDE & MATE went perfectly together.
I also found the order they are installed matters; I played with what I wanted (decided to drop KDE & just use MATE; if re-doing today I'd go the other way & keep KDE)
On the KDE/MATE clash, I couldn't find the clash (to raise bug report) as I loaded KDE/MATE combinations easily in testing with any other base desktop; the issue only occurred with my wanted desktops... ie. the more you have the more problems you should expect.
I use many boxes, and nearly all have multiple desktops; I'd expect very minor problems only when adding >2, and more as the number increases.