How many times has someone other than Bruce Wayne had to assume the mantle of Batman?

Question

How many times has Bruce Wayne died, or at least been injured or incapacitated to the point where someone else has had to step in and assume the mantle of Batman?

I know during the Knightfall arc, Bane breaks Bruce Wayne’s back and while he is recovering, Jean-Paul Valley takes his place as Batman.

How many other characters have taken Bruce Wayne’s place?

Answer 1

In the pre-New 52 Earth-1 continuity, there have been a couple of alternate Batmen.

• Jean-Paul Valley, better known as Azrael, was mentioned in the question. Bruce tapped him to take up the Cowl following his defeat at the hands of Bane in Knightfall, but Valley went slightly peculiar stark-raving mad, and was deposed in the sequel series KnightsEnd
• After stripping Valley of the title, Bruce passed it to Dick Grayson, setting off the Prodigal storyline, which ran for about a year. Dick took up the mantle again following 2009's Battle for the Cowl arc
• Batman #700 gives us a glimpse at a couple of future Batmen, who have presumably taken over for Bruce in his old age; the notable ones are:
  • Dick Grayson, again
  • Damian Wayne, Bruce's son by Talia al'Ghul
  • Terry McGinnis, best-known from the DCAU show Batman Beyond, in his comics debut
• Damian Wayne is also shown to have taken up the cowl in a hellscape future in Batman #666, following Bruce's death
• Tim Drake as been shown in various alternate futures to take over from Batman, notably in the Teen Titans: Titans Tomorrow storylines. He also swaps capes with Bruce in the frankly bizarre Young Justice: Sins of Youth storyline, where Klarion the Witch-Boy magically ages Tim and de-ages Bruce
• You may also count Jason Todd, who assumed the Cowl (without permission) during the Battle of the Cowl storyline. He was a rather brutal Batman, as you'd expect, but he's deposed by Dick by the end of the arc
• An interesting edge case: during the "Blind Justice" storyline, Bruce is temporarily rendered paraplegic and is understandably unable to be Batman until he recovers. To maintain order, Bruce remotely operates Roy Kane, a character introduced for (and killed during) this storyline, using some macguffin mind-controlling technology¹

In the New 52 continuity, currently only one other person has taken up the Cowl: Commissioner Gordon, who used a Batman-shaped mech suit to carry on the legacy after the real Batman was presumed dead².

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¹ Thanks to FuzzyBoots for pointing this out in comments

² Nod to Durakken for reminding me of this in comments

Answer 2

How many other characters have taken Bruce Wayne’s place?

For one without Batman's consent, in the story I Am the Batman (Detective Comics # 472, September 1977) Hugo Strange impersonated both Batman and - for a prolonged period - Bruce Wayne (emptying Wayne's bank accounts in the process), whilst Batman was a prisoner at Strange's 'clinic'!