ANSWER: 온가지 is the original form of 온갖, which means "all (온) kinds of (가지)."
One interesting Korean spelling rule is that if a word ends in a vowel, sometimes that vowel can be dropped, and the remaining consonant in front of it can be used as the final consonant of the syllable just before it. So, in the case of 온가지, the vowel "ㅣ" in 지 is dropped, leaving just ㅈ, which is then added to the syllable 가 just before it, forming 갖. That means that 온가지 becomes 온갖.
Another example is 엊그저께, which originally was a combination of 어제 (yesterday) and 그저께 (the day before yesterday). So, the 엊 part of 엊그저께 is a result of the ㅔ in 어제 being dropped, leaving just ㅈ, which is then added to the 어 to form 엊, and that results in 어제그저께 becoming 엊그저께, which means "a few days ago."
