Translation:Where is the closest convenience store?
Is ichiban commonly used in this context, like "closest" rather than very best?
一番（いちばん） literally means number one （一: one, 番: number), so in this context, it's 'the number one close convenience store'. With すきな, it's 'the number one like' (literally).
So actually, いちばん can be used in lots of places, since it is just turning the adjective it's attached to, into the superlative form of the adjective.
I understand the context that it is being used in in this sentence, I was just wondering if this was common usage among native speakers. For some reason, I have the (probably wrong) impression that ichiban would be more commonly used for things that are being described favorably.
I live in Japan and it is commonly used. "most" doesn't really in Japanese. You use either "とても/めっちゃ" (second one being oral language so more casual) or if you want to emphasize the fact that you don't know something more "adjective" than that : いちばん
It would be more natural to say "一番近いコンビニに行きたいんですが" - literally "I want to go to the closest convenience store...". As I understand it, the phrase どこですか implies that you expect the listener to know the answer and help you. But if you just say, "I'm trying to get to X", people will both understand what you're really asking, and consider it less rude.
This is not about different definition, but just the use of "number one" that is different. Japanese don't say "most" so they will say "number one". :) But yeah, ichiban has a lot of uses.