How can I increase my experience gain when handing in quests?

Yes, using Mentats is just like having two extra Intelligence points for extra experience when handing in a quest. However, you can't stack more than one Mentats effect at a time (two extra points is the max).

Apparel

If someone could confirm this in the comments below, I'd appreciate it. I think the easiest place to find the hat+body items is the Quincy Ruins. You can head there at the beginning of the game straight past Diamond City. In the middle of the town (held by raiders) is a storefront (Guns, guns, guns, it's called) that has a bunch of clothing items on an open shelf. One of them is the Ushanka Hat. I also went to the top floor of this building and found a Vault-Tec Lab Coat in the dresser next to the bed. (Can someone confirm whether this is random or a set item?)

  • Vault-Tec Lab Coat (+2 INT – locations)
  • Press Hat (+1 INT – trade with Piper when she becomes your companion)
    • Alternative: Ushanka Hat (+1 INT – Quincy Ruins, see above)
  • Liam's Glasses (+2 INT – at end of Plugging a Leak quest late in the game)
    • Alternative: Road Goggles (+1 INT – locations)

These items add to the Intelligence stat for extra exp gain atop Mentats when handing in quests. 10 (maximum base INT) +5 (apparel additional INT) +2 (temporary Mentats buff) = 17 INT points!

Sleep

Waking from sleep with a "well rested" notification denotes a 10% experience buff for 12 hours (though I haven't been able to confirm this myself). If you sleep with a romanced partner, the bonus increases to 15% ("Lover's Embrace").


In addition to the above menioned items I want to explain more about the idiot savant perk. Basically It is always a good pick till level 2. Why? Easy! There are two ways to boost your XP gain on a int/savant basis.

First one is INT 2 and idiot savant 2. The outcome is an avarage XP boost of 1.484. With each further INT level the chance to proc the idiot savant drops. Your XP boost drops, because the additional XP due to more INT does not overcome the lost XP due to the lower idiot savant proc probability. But luckily that changes after you get INT to a higher level.

Second one INT 15 and idiot savant 2. At INT 15 you gain more XP than with the first method. The probability of an idiot savant proc is 1% if you reach INT 10 and above. But the bonus XP due to your INT overcompensates that after you reach int 15. You have a boost of 1.508 now (or 1.45 without idiot savant, which means you need int 17 now to get 1.51).

Source: http://fallout.wikia.com/wiki/Idiot_Savant

PS: When getting XP for a quest and the perk idiot savant is skilled: Quicksave and turn in the quest. Idiot savant can proc for a quest reward! Did not proc? Reload and reload and reaload... till profit!


Further to Wolf's accepted answer, there are a couple of extra options to increase XP gain, and even more with DLCs. Also, there is an important note regarding Liam's glasses.

Maximum base INT is 12, not 10:

  • once your INT is 10, decrease it using alcohol and then use "You're SPECIAL!" book
  • finally collect the bobblehead

Apparel:

  • Liam's glasses give +2 INT - they are however quite far down the main storyline, require to not join the Railroad (if you completed Tradecraft quest, Plugging a Leak won't be available) and are exclusive with Ballistic Weave and Institute Killer Weave clothing mods. Deliverer pistol can still be obtained by doing Tradecraft partially
  • Destroyer's Helmet gives +1 INT - sold by Penny Fitzgerald in Covenant
  • Far Harbour DLC: The Captain's Hat gives +2 INT
  • Lab Coat outfit (cannot be worn under armor) gives +2 INT - note that there is about a dozen of variations like Vault Lab Coat, Institute Lab Coat, Dirty Lab Coat etc.
  • Mechanic Jumpsuit outfit gives +1 INT
  • legendary armor pieces with Sharp prefix give +1 INT - they cannot be worn over Lab Coat but they would give +5 INT instead of +2 INT (set difficulty to highest to increase the number of legendary enemies spawning)
  • Nuka World DLC: Fancy Suit & Tie gives +1 INT and can be worn under armor
  • Creation Club: Military Backpack with Institute patch gives +1 INT
  • Nuka World DLC: Lucky Rabbit's Foot gives +10% XP when equipped - it is a grenade which does nothing when thrown, apparently it has a very, very low chance of dropping from any Nuka World raider

Power armor:

  • helmet's Internal Database misc mod gives +2 INT

Consumables:

  • Berry Mentants give +5 INT - doesn't stack with Mentats
  • X-cell gives +2 INT - stacks with Mentants, however it is quite rare
  • Squirrel Stew gives +2% XP for 2 hours (I think these are real time hours)

Perks:

  • Piper's companion perk doubles XP for persuasion and location discovery
  • Nuka World DLC: Porter Gage's companion perk gives +5% XP for kills
  • Idiot Savant (req. LCK 5) gives a chance of gaining x3 or x5 XP
  • Night Person (req. PER 6) gives +2 or +3 INT at night
  • Well Rested / Lover's Embrace

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