This is more of an aesthetic / rhythmic issue than a grammatical one. Technically you can keep adding 的 to make a chain as long as you like, but it won't be "nice". It's like saying "that" repeatedly in English, e.g. "the man that ate a dog that ate a fly that watched a cat that liked to browse stack exchange." Grammatically valid but not natural.
The general rule with multiple 的s is to only keep the last one, but sometimes you do have to use multiple 的s. It's usually worth looking at rephrasing the sentence if possible to avoid this (in writing), but in speech I would say just don't worry about it. Even native speakers say things that don't sound perfect, especially if they pause for thought in the middle of an utterance, e.g. "妈妈的那个白色的狗" isn't nice Chinese but people could easily end up saying it.
Also, there is a slight difference between 白狗 and 白色的狗. 白狗 sounds like a particular established category of thing, 'white dog', whereas 白色的狗 is modifying an existing category of thing, 'dog that is white' (although you would translate it into 'white dog'). This may seem obvious but there is a subtle but important difference.