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Would there be any damage to a laptop or another electronic equipment if I use the internal battery supply till it has been completely depleted, then I recharge it to maximum, then wait for it to become completely depleted again, and then repeat.
If so, what would be some of the long term effects?
Batteries have a limited lifetime (including sitting on a shelf) due to ageing and also a limited number of charge/discharge cycles, which depends on the average depth of discharge (at least for lithium ones). Example:
Lithium batteries don't suffer memory effects (forcing you to fully discharge regularly) so it's actually preferrable to discharge it partially like the graph suggests.
I can only tell about users concerns though, I know how batteries work but not enough to say why exactly this happens (that's when experts on the topic come in).
Anyway, the answer to your question is: your battery won't last long. The undervoltage protection circuit will cut off each discharge before it instantly kills the battery, but it will be dead in very little time regardless (800 cycles judging from the graph for that particular battery, or about 2 years if fully depleted and recharged every day).
Most rechargeable batteries, particularly LiOn, have a lifetime defined by number of full discharge/recharge cycles. For most LiOn batteries, this is around 500 cycles. Therefore, if you cycled AC to the laptop 5 times per day, the battery would be dead in about 100 days. (BTW, partial cycles count much less against the battery lifetime.)
On the other hand, it is good to fully cycle a laptop battery around once per month. This allows the battery level monitoring circuitry to "re-synchronize" with the actual battery level.
So to maximize battery life, you should run the laptop on AC as much as possible, while occasionally running on battery for a full cycle.
I want to add a bit about why this isn't the best idea for your battery's health from the electrochemistry side of this question.
Lithium ion batteries have a cathode - a layered structure with some lithium in it - and an anode - usually carbon. You charge them by moving lithium ions into the layers of the cathode, and you discharge them by pulling the lithium back out (figure from Sigma Aldrich):
The big problem now is the cathode. If you discharge a battery to extremely low voltages, the layers of the cathode don't have anything to keep them apart, and the layered material starts breaking down. This causes irreparable damage to the battery - it loses the ability to hold charge and becomes a battery-shaped paperweight. Your laptop probably stops you way above the point-of-no-return, but it's still better for your battery to keep it from getting fully discharged.
Aside from this, cycling a battery causes it to lose capacity for multiple reasons. This is covered in the other answers.
