Why it exists. For something that keeps changing: 越来越yuè lái yuè + an adjective, or a feeling verb like 喜欢xǐhuan, 爱ài, 想xiǎng. Mark in Text 3: 我手机的电越来越少了。Wǒ shǒujī de diàn yuè lái yuè shǎo le.

Something越来越adjective / feeling verb(了)

+我们的小区越来越干净了。Wǒmen de xiǎoqū yuè lái yuè gānjìng le.Our estate is getting cleaner and cleaner.

+我的手机越来越慢了。Wǒ de shǒujī yuè lái yuè màn le.My phone is getting slower and slower.

+马克越来越会做饭了。Mǎkè yuè lái yuè huì zuò fàn le.Mark is getting better and better at cooking.

+到了十一月，我越来越想家了。Dàole shíyī yuè, wǒ yuè lái yuè xiǎng jiā le.Now that it's November, I miss home more and more.

Key points

No 很hěn, 非常fēicháng or 太tài before the adjective: 越来越慢yuè lái yuè màn, not 越来越很慢yuè lái yuè hěn màn. 越来越yuè lái yuè already says how much.
了le at the end is common: it shows things have changed (and are still changing). 我的手机越来越慢了wǒ de shǒujī yuè lái yuè màn le.
Mostly adjectives and feeling verbs: 越来越喜欢yuè lái yuè xǐhuan, 越来越想家yuè lái yuè xiǎng jiā. A plain action like 学xué doesn't work on its own, but a verb with 会huì or 有yǒu does: 越来越会做饭yuè lái yuè huì zuò fàn, 越来越有意思yuè lái yuè yǒu yìsi.
越来越不…yuè lái yuè bù… works too: 我越来越不想下车了wǒ yuè lái yuè bù xiǎng xià chē le (Lisa in Text 4).

Common mistakes天越来越很冷了。 → 天越来越冷了。Tiān yuè lái yuè lěng le. No 很hěn after 越来越yuè lái yuè.
我越来越学汉语。 → 我越来越喜欢学汉语了。Wǒ yuè lái yuè xǐhuan xué Hànyǔ le. 越来越yuè lái yuè goes with adjectives and feeling verbs, not with an action like 学xué.
他非常越来越累。 → 他越来越累了。Tā yuè lái yuè lèi le. No 非常fēicháng either: 越来越yuè lái yuè already says how much.