Combining two verbs without て


So I was reading the sentence, 今度遊びに来てください. And I believe you intepret that it
means like, next time you should come and hang out, To break it down, I
believe に is something like "going" or "to". So you are coming to do
something,but I'm wondering is why is 遊び written here?

Could you say, 遊んで来てください. That technially means play and come? But then if you
flipped it, 来て遊んでください. That means come and play, so that seems pretty similar
to the first sentence.

TLDR: why is the format 遊び used in the sentence, and could you also use て form
to connect the two verbs?




～にいく／～にくる (where ～ is the -masu stem of a verb) and ～ていく／～てくる are both
standard constructions, and there is some overlap in their usage.

If you say ～にいく then it implies you are going to a specific place in order to
do ～. For example, 公園に遊びにいく means "I am going to the park to play". So in your
case, 今度遊びに来てください means "Please come (here) to play next time", i.e. it's an
invitation to come back to play with the speaker.

On the other hand, ～ていく has a few different interpretations depending on
context, but one main interpretation is that the subject is going somewhere,
and doing ～ before or as part of the journey. For example, 歩いていきます means
you're walking somewhere (you're going there, and walking is part of the
journey). ～てくる is similar, but it implies that the end of the trip involves
coming back home or to coming to the person being addressed. For example, when
you leave home you will say 行ってきます, meaning (roughly) "I'm going out, and will
be coming back". You could also say だいどころからいすをとってきます to mean you're going to
bring a chair from the kitchen.

In that context, 遊んできてください sounds like an invitation to go away, play, and
then come back, which sounds more like what a frustrated parent might tell
their hyperactive child.

