I know that you can always ask your companions how your relationship is going but when I'm trying to max everyone's relationships, it'd be nice to know exactly how much my companions like me. Is there an actual value for this? Can I tell how close to the next relationship rank I am?
From the Fallout Wiki
The affinity value is a hidden statistic, meaning it is not possible to check the progress without console commands.
If you're not on PC, it appears you're out of luck as there is no way to know. If you're on PC, you can check through console commands.
Open up console
Click on the companion you wish to check
Here is a scale for telling how much affinity is earned for each action:
Companion Liked that +15
Companion Loved that +35
Companion Disliked that -15
Companion Hated that -35
If you're looking to trigger new quests from companions, the Wiki also says:
For most companions, new dialogue will be triggered when the affinity value reaches 250, 500, 750 and 1000.
Some extra information about affinity:
Typically, all companions begin at an affinity value of 0. If the affinity value reaches 500, they will "admire" the Sole Survivor and often reveal new dialogue. Raising their affinity to 1000+ (max is 1100) will result in them "idolizing" the Sole Survivor, at which point the Sole Survivor are granted their companion perk.