Would deploying flaps increase the maximum altitude an aircraft could reach?

No. The absolute ceiling of an aircraft is primarily a function of thrust vs drag. Deploying flaps increases drag, so even if lift were to increase with a small application of flaps, without additional thrust the gain would only be temporary and the absolute ceiling would actually be less.

No, it will reduce it.,

Flaps increase lift but at the expense of an even larger increase in drag in proportion.

At take off and landing, the extra drag is less of an issue since you are operating at reduced velocities, but at altitude, a plane must fly as fast as possible to get the most lift. The ceiling is reached when the aircraft is getting the highest possible lift for a drag that is equal to the highest possible thrust at that highest possible speed.

Maximum altitude for extending flaps on a Boeing (any type) is 20,000 ft -

For the DC8 - flaps max speed (VFE) or Mach number is 230 KIAS or Mach.46 -

Would be impossible to extend flaps and be below 230/.46 at cruising altitude -

You would stall...

Your cruising altitude and speed margins are limited by the "Coffin Corner" -

And the weight at which you are cruising -

In other terms, remain between Vmo and Vs -

A heavy 747 can barely fly at 30,000 ft - yet if light, it could cruise at 45.000 ft -

Check that on Wikipedia -

What you are trying to say is to do like TWA 727 -

Extend trailing edge "flaps 2", and not extend leading edge flaps or slats -

See what happened to that airplane...!

http://en.wikipedia.org/wiki/TWA_Flight_841_(1979)

When I was F/O or F/E 727 in the 1970's, there were a few captains who did that -

We were lucky we never ended practicing snap rolls -

No, the increased drag would counter any benefit of increased lift.

NO !!

That would slow the aircraft down and it would start to lose height.

whats your thinking behind this real odd question. leaving this one alone

No.