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The Architects of Agony and Hyperstimm Backlash stratagems are worded as follows:

Hyperstimm Backlash wrote:Use this stratagem at the start of the battle round. Choose a unit from your army that has the Combat Drugs ability. Until the end of the battle round the bonus that unit receives from its Combat Drugs is doubled. At the end of the battle round, roll a d6 for each model in that unit. For each roll of 1, that unit suffers a mortal wound. If you use this Stratagem on a model that has the Stimm Addict Warlord Trait and/or the Phial Boquet Artefact of Cruelty, all its bonuses are doubled, but it automatically suffers D3 mortal wounds at the end of the battle round.

Architects of Pain wrote:Use this stratagem at the start of the battle round. Choose a unit from your army that has the Power From Pain ability. Until the end of the battle round the bonus that unit treats the battle round as being 1 higher than it actually is when determining what bonuses it gains from the Power From Pain table. This is cumulative with other, similar effects (such as the Kabal of the Black Heart's 'Thirst for Power' obsession.)

Can these stratagems be used multiple times per battle round?

Hmm they arent listed as one use only.. they don't take part in a phase so the matched play restriction doesn't apply.. looks like yes.

That's what I thought, just double checking if I am missing anything. Next question is can they be used repeatedly on the same unit? If so does that do anything?

I would say yes, you can, because the Start of the Battle Round is not the same as the start of the Movement Phase, there just isn't an actual phase between them, but there is a moment in time between them.

What do you think happens if you use them multiple times on the same unit?

Drager wrote:
What do you think happens if you use them multiple times on the same unit?

The rulebook FAQ states that Identical Auras don't stack, but these aren't auras, that being said...

As for the doubling stratagem, it wouldn't work because it only doubles the original effect, not the bonuses effect. You've basically got two rules saying "double the basic effect", but that doesn't make it 4x the effect because it "overwrites" the first.

Order of operations:
Drug says do this
Stratagem changes drug to say "Do this but DOUBLED!"
Stratagem changes drug to instead say "AH-HA! THE FUN HAS BEEN DOUBLED!"

So the result is still a doubled drug but more pointless overdose wounds.

Same for the Power From Pain stratagem. You have two rules saying "Treat the battle round as one higher", but they apply to the current battle round number, not to the altered number from the first use of the stratagem.

The Architects of Pain says it stacks with other similar effects, would it not be a similar effect to itself?

I agree on Hyperstimm Backlash, you'd just take extra mortal wounds for no reason.

Drager wrote:
The Architects of Pain says it stacks with other similar effects, would it not be a similar effect to itself?

I agree on Hyperstimm Backlash, you'd just take extra mortal wounds for no reason.

The key word there is "other".

So the reapplication of the same effect is still the same effect, being a separate instance doesn't make it distinct? Agreed.