We know that the faster we move the more time gets dilated. And we have a limit of speed in physics which is the speed of light and nothing can travel faster than light.
"So the greatest extent to which time can get dilated (according to the speed concept, not gravity) is the time dilation for a system travelling at the speed of light right?"
Photons do not feel time the way everything else does. For them time is not a dimension. If we we were to travel at the speed of light, we will lose all sense of time, it would be like standing in empty room without any natural indications of time. Considering the time dilation formula
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which means time would be infinitely dilated that is we won't be moving across time at all, thereby losing any sense of the presence of dimension called time.
This is a topic which has sparked centuries of debate and discussion, thereby there is no perfect solution to this, only until one actually travels at the speed of light.
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