Is it possible to be reborn for all eternity?

SN 22.99 - Gaddula Sutta: The Leash

http://www.accesstoinsight.org/tipitaka/sn/sn22/sn22.099.than.html

Supreme Buddha’s answer to your question in found in the Gaddula Sutta. So it is more or less eternity as He says, “Unimaginable, Bhikkhus, is a beginning to the round of births [and deaths]. For beings obstructed by ignorance and fettered by craving, migrating and going the round of births, a starting point is not evident.” Only if you could realize the Dhamma of a Supreme Buddha, you will escape this eternity. But you know what, it is so rare is it to find a Supreme Buddha! Do you have any idea what an aeon (world cycle / kappa) is? It is equivalent to eternity. Only by a simile you will get an idea of it. The Buddha offers a vivid simile to suggest the eon's duration.

Anamatagga-samyutta — The unimaginable beginnings of samsara. In Samyutta ii, Chapter XV, the Buddha used the parables of the hill and mustard-seed for comparison:

• Suppose there was a solid mass, of rock or hill, one yojana (eight miles) wide, one yojana across and one yojana high and every hundred years, a man was to stroke it once with a piece of silk. That mass of rock would be worn away and ended sooner than would an aeon.

• Suppose there was a city of iron walls, one yojana in length, one yojana in width, one yojana high and filled with mustard-seeds to the brim. There-from a man was to take out every hundred years a mustard-seed. That great pile of mustard-seed would be emptied and ended sooner than would an aeon.

In the last 91 Kalpa (Aeons) only 7 Buddha’s have come. At the time of the next Buddha, the average lifespan of a human is going to be eighty thousand years. (In the present day it is around 60 -80 years). What are our chances of being born during that time, when we could never meet any of the past Buddhas and are still in this samsara?

A person with achieved enlightenment will end the birthing cycle and go to nirwana. It is the greatest achievement of a Buddhist. Lord Buddha has told until there is no beginning or an end to the birthing cycle of all beings. So it is wise to go to nirwana and end the tormented cycle of sansara.

From the viewpoint of most (if not all) Buddhist schools, you have been being reborn since beginningless time- all the way through repeated universes as they come and go in their own cycles.

You will continue to do so.

The point is, you may be fed up of the suffering that you continue to experience. Enlightenment (or Nirvana) is a permanent (as in forever) and irreversible step out of the suffering.

In some Mahayana schools, this does not mean an end to the continuum of consciousness though. It's not really 'rebirth' as you think of it, as being free from the idea of 'self' frees you from having to plod around as a single being. Once enlightened a Buddha is able to be reborn (or merely manifest) simultaneously in multiple (no limits) places across the universe.

Many people are interested in living forever (eg Christians) because such people do not think living is suffering. Instead, they wish to spend forever with their loved ones. That is why they are not queuing up to practise Buddhism.

My impression is Buddhists are similar, in that they wish to believe in reincarnation so they can live more lives. That is why most Buddhists are not practising the path, hardcore, but preaching that Buddhists must believe in reincarnation, rebirth, whatever.

Also, Buddhism does not state making merit ends suffering (SN 12.51) so what is its relevance?

Also, Buddhism does not state all beings are destined for enlightenment (AN 10.95).

Imo, the questions are an extreme & pointless misrepresentation of Buddhism. If the questioner truly believe they were suffering, they would be a monk urgently putting out the suffering, like as if their hair was on fire.