So, この時、あまりたくさん置きすぎないようにして下さい can be deconstructed as follows:
置き - put in, fill
たくさん置き - put in many, fill it up
置きすぎ - overfill
- don't know whether it's been overfilled by a small or large amount; just that it's in excess of the expected limit
たくさん置きすぎ - overfilled it {a lot / with too many}
- definitely know it's been overfilled by a large amount
あまりたくさん置きすぎない - not overfill it very much / with so many
- don't overfill it to such an extent / in excess of the limit, but still make sure its the full amount
In general speak, it's a bit like a glass of wine in a restaurant. Although the standard amount is 100ml, the customer wants you to fill their glass right up to the top (overfill it) but, not to the excessive degree / extent that the wine's spilling over everywhere on their table.
Related Expressions / Idiomatic Usage:
- できるだけ飲みすぎないようにしてください
- 顔がたくさんすぎて気持ち悪い
- 君に伝えたいことがたくさんありすぎて
- たくさんありすぎる
- 沢山すぎて
- たくさん作りすぎたソファ
There are a few points to note here:
- たくさん = lots of, many, much
- 過ぎる = too much / many of N, or over-V
- あまり~Sない = not very much / many of ~
たくさん is generally used to indicate the amount of something, using
- an adverb: 今年はたくさん雪が降った。A lot of snow fell this year.
- a noun: この部屋には机がたくさんある。 There are a lot of desks in this room.
過ぎる can be interpreted two ways in this context - put in too many nouns or over-verb. Overfill (over-put) works better as it works better with the amount and still enables the amari~nai pattern to indicate overfill but not to excess or immoderately. As such, and it avoids the double adverb problem you mentioned - 'too too much'.
あまり~Sない is an adverbial pattern that is used with a negative predicate to indicate the degree or extent of something
- この本はあまりよくない。 This book isn't very good.
- 鈴木さんはあまり食べない。 Suzuki doesn't eat much.
- 私はあまり速く走りません。 I can't run very fast.