I have been playing this song that uses a very strange arpeggio in it. It has a major third and a minor third in it. I have no idea what to call it and most the names I'd imagine it having (A#Mb3) seem rather contradictory... well here it is so you can decide.

The rhythm guitar plays octaves behind it so I suppose the major/minor doesn't really matter but it will bother me if I don't find out the name. Plus it might be fun to make a song that blurs the line between major and minor... some how!
Bb minor add 10 is what I came up with, but I'm probably wrong.
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Theory is descriptive, not prescriptive.

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theory states 1+1=2 sometimes in music 1+1=3.
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I've never heard of that chord before.
Why over complicate it? You will never, to my knowledge, have a chord containing both "a minor third and major third" in which both actually function as thirds. In fact, any time you have both intervals, it's almost guaranteed that what you are dealing with is a major chord. The "minor third" is better though of just as some added chromatic tone, or, if it occurs over a dominant chord, a #9 and suggestive of an alt5ered dominant chord.
Someones knowledge of guitar companies spelling determines what amps you can own. Really smart people can own things like Framus because they sound like they might be spelled with a "y" but they aren't.
^ yes. Except in this case it is not an altered dominant extension but an add chord cause there is no 7th. But othewise yes. Just as when there is a third and a fourth present the third is called an 11th - when there is a 2nd and third present the 2nd is referred to as a 9th. When there is a third and flat third present in the chord you could also call it a third and sharp second. Since there is a third and second you would call the second a ninth. So a Major chord with an added #9
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Bb C# F Bb D F

Your song or someone else's?
If latter, how about a title.
What does the bass do? What comes before and after?
Analysing chords in isolation is pointless exercise as it doesn't consider their FUNCTION
It could perhaps be built on an F# and include a minor 13 - but who knows until we have all relevant information.
It's a Bb(add#9), as 20Tigers said.
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It's a Bb(add#9), as 20Tigers said.

Highly doubtful that the 3rd would be voiced above the augmented ninth, therefore IMO that solution is unlikely. But as I stated before, no definitive answer can be given without knowing the context in which the chord appears.