I was reading the description of sickening radiance and was wondering if there were any other ways to give another creature levels of exhaustion in combat? It seems like it could be very DM dependent.
At the time of writing this I can find no character or monster ability (in the PHB, DMG, MM or XGtE) besides the spell you mention that directly causes exhaustion. There are two indirect, but canonical ways that could work in the timescale of a combat:
The fastest way to gain exhaustion listed in the DMG is to sprint:
During the chase, a participant can freely use the Dash action a number of times equal to 3 + its Constitution modifier. Each additional Dash action it takes during the chase requires the creature to succeed on a DC 10 Constitution check at the end of its turn or gain one level of exhaustion. (p.252)
So if you give them reason to run and then chase them, you can indirectly cause them to gain exhaustion.
There is at least one feature that causes the user to gain exhaustion, the frenzied rage of the barbarian. If you dominate such a creature (eg. with the spell dominate person), you can cause them to use the feature and thus force exhaustion upon them.
With the release of Explorer’s Guide to Wildemount, there is a consumable that forces saving throws against exhaustion, soothsalts:
For each dose of soothsalts consumed, the creature must succeed on a DC 15 Constitution saving throw or gain one level of exhaustion—an effect which is cumulative with multiple doses.
Any way you are able to make someone drink something, whether through force or deception, you can force exhaustion with soothsalts - and it becomes easier with each dose as the target takes more levels of exhaustion.
This isn't in combat, but since it can be done as a straightforward preparation for a specific battle, I'll write this as an answer:
It is possible to achieve a known enemy to have 1 level of exhaustion before combat via Dream spell:
and then the target must make a Wisdom saving throw. On a failed save, [...] prevents the target from gaining any benefit from that rest.
To be sure, you have to attack within 8 hours of Dream ending, to prevent target from sleeping more (unless you have multiple casters with Dream to torment target for a longer time). Spell also does psychic damage, so it is best to attack before target has a chance to get healed. And as a bonus target does not gain spells etc long rest benefits, either.