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Let's assume we have Warlock 10 / HellFire Warlock 1. He has following stats

  1. Eldritch Blast base damage 6d6
  2. He has equipped Greater Chasuble of Fell Power (+2d6 to Eldritch Blast)
  3. Uses Gloves of Eldritch Admixture or Warlock's Scepter (for +4d6 to Eldritch Blast) as swift action
  4. Uses Eldritch Essence Brimstone Blast (essence is added to Eldritch Blast that deals 2d6 fire damage each round for 2 rounds)
  5. Uses Hellfire Blast (Hellfire Warlock Feature) for 2d6 damage
  6. Uses feat Maximize Spell-Like ability he has Applied to Eldritch Blast

What total damage he should expect, assuming he hits target that fails Reflex saving throw and has no energy resistance etc? What parts of this damage would benefit from Maximize Spell-Like ability feat and why?

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Maximize spell-like ability reads:

All variable, numeric effects of the spell-like ability are maximized, dealing maximum damage, curing the maximum number of hit points, affecting the maximum number of targets, and so on.

The only exception listed is this:

An empowered maximized spell-like ability gains the benefit of each feat separately (getting the maximum result plus one-half the normally rolled result).

Since all of your listed buffs are listed as "increase the damage of your eldritch blast by +Nd6" which is simply making your blast do more damage instead of dealing bonus damage as a separate effect, and since brimstone blast (and vitriolic blast for that matter) both do random damage that is considered part of the eldritch blast spell-like ability, then a maximized eldritch blast will work fully on the entirety of the listed damage. Note that Mortalbane from the Book of Vile Darkness would also be maximized since it, too, increases the damage of the ability by a random number rather than doing additional damage as a separate effect when you deal damage with the ability.

The total damage you would expect is:

36+12+24+36+12=120

Note that brimstone blast deals burn damage on the first turn, too, so it does 3 ticks total, not 2.

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