I have been playing a lot today.
I have played a lot today.
Can either be used if the day isn't over and I might continue playing? Or does the present perfect mean I have stopped playing for the day?
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Saying "I have played a lot today" with a tone suggesting that you did not expect to play so much or you played more than how much you normally play could indicate that you don't intend to play anymore.
Without the tone, both the statements do not give any hint of whether you will play later in the day.