I wonder the kind of effect that eating many Goodberry in one day can do to a character.

The spell description states:

... the berry provides enough nourishment to sustain a creature for one day.

But, there is no additional explanation of what "enough nourishment" means, or what the consequences are if you nourish yourself more than what is necessary in a day.

There is some elements in this answer, and also in the comments, but I want to know if weight is the only possible drawback (I mean, a character could also train more to compensate for the surplus intake calories)

There are no other effects (unless your DM rules there to be)

(besides the 1HP you get from each berry)

Goodberry specifies all of its effects in the spell description. Nowhere does it state any effects that come about from eating many of them. And spells do only what they say in their description and anything further is the realm of a DM to decide. So, by the rules, goodberries have no additional effects on you even when eaten many at a time.

There are no rules in the books that even state what would happen if you ate too much normal food over a long period of time (though we could certainly extrapolate in that case), much less magical food.

Should a DM rule that there are extra effects?

This really depends on the table. There could be an some fun in saying that a character starts turning red or gaining weight¹ or any other effect a DM can imagine for these eating the berries, but, in the end it should be up to that DM to decide what is the most fun for their table.

It is certainly extremely arguable that magical nourishment works in any way like normal food-eating, but that is a detail that each DM can decide for themselves.

I would generally advise if a DM was to go down this route to only introduce mechanically meaningless effects and definitely not ones that impose any kind of negative mechanical effect.

^{1 - Note that playing up a weight gain for laughs is also not cool at many tables, so know your table.}