I have a Half Orc who is proficient in Investigation so I was wondering, can I take the Investigator feat and also the Prodigy feat to gain triple proficiency on Investigation checks?
If I become level 20 with this, will I have a plus 18 to all investigation roles so the minimum would be nineteen on any roll?
The rule on proficiency bonuses states:
If a circumstance suggests that your proficiency bonus applies more than once to the same roll, you still add it only once and multiply or divide it only once.
So no matter how many features you have that double your proficiency bonus, you only get to multiply it once.
The Investigator feat (from Unearthed Arcana: Feats for Skills) either gives you proficiency in Investigation, or doubles your proficiency bonus for Investigation checks if you were already proficient.
However, the Prodigy feat (Xanathar's Guide to Everything, p. 75) specifies:
Choose one skill in which you have proficiency. You gain expertise with that skill, which means your proficiency bonus is doubled for any ability check you make with it. The skill you choose must be one that isn’t already benefiting from a feature, such as Expertise, that doubles your proficiency bonus.
(Emphasis mine.)
So I'm afraid you cannot also pick Investigation for the Prodigy feat if you already have the Investigation feat.
Even if you were to choose the feats in the other order (Prodigy first, then Investigator), you would only add double your proficiency bonus for the reasons noted in Theik's answer. Plus both features only state that your proficiency bonus is doubled for the check, referring to your actual proficiency bonus (i.e. if your bonus is +2 then you add +4; nothing implies that you triple anything or double the already doubled number)
Even the caveat in Xanathar's is there just to save people from overhinking thigs, but is irrelevant. Both features say the same text:
Choose one skill in which you have proficiency. You gain expertise with that skill, which means your proficiency bonus is doubled for any ability check you make with it.
Or the other feat:
You gain proficiency in the Investigation skill. If you are already proficient in the skill, you add double your proficiency bonus to checks you make with it.
Even the rogue feature:
At 1st level, choose two of your skill proficiencies, or one of your skill proficiencies and your proficiency with thieves’ tools. Your proficiency bonus is doubled for any ability check you make that uses either of the chosen proficiencies.
You can pick them, but all of them do the same thing. They set the value added to the skill check from proficiency to double. You can pick and choose what doubling you want to use, but in the end it is just that. double.
It is the same thing as trying to get five +2 longswords to get a +10 to hit and to damage. You use one or the other, they don't stack.
Also analogous is the AC from wearing armor or other effects. Like the barbarian. Their base AC is either 10+CON or the AC value of armor. Do not stack.
