Which brass wind instrument has the widest range?
I mean, which one has more notes between playable lowest note and playable highest note?
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The range of any brass instrument depends on the player. There are trombone players who can play six octaves. Whether the extreme notes are musically useful is a matter of opinion.
In general, the larger and lower instruments in each family tend to have the widest range. As you make an instrument larger, it gains range on the bottom faster than it loses range on the top.
The tuba has a range of about four octaves. For reference, Wikipedia lists ranges for all instruments (Here's the tuba page), although these tend to go to the absolute extremes, beyond typical idiomatic writing. Those lowest notes are very indistinct and can't be used melodically, and the notes above the bass clef staff are rarely used outside of solo works.
The horn (a.k.a. French horn) also has a total range of about four octaves. Despite being thought of as a higher instrument (between the trumpet and trombone) it actually has the tubing length of an F tuba and thus the potential to play very low. However, most players aren't capable of playing well across the full range, so the effective range is somewhat less.
As Carl Witthoft comments on your question: theoretically there is no limit for overtones blowing in any tube.
All Brass instruments have the same range: it is usually the series of the overtones: 2 octaves + the 6 semitones below the root tone reached by the valves 123 (notated F#) and ca. a 5th up the notated high C, but these tones are depending like the other answer says from the individual abilities and technique of the brass player.
Now there are some instruments with an additional valve (Euphonium, Eb-Bass, certain Trombones) with a fourth valve that enlarges the pitch range by 4th. These instruments are the range record breakers.
My answer concerns the practice of the brass instruments: that means I am excluding the pedal tone that is rarely used (maybe as joke - as I didn't, for demonstrations and eventually futuristic compositions). To the pedal tone we can also count downward the 2,1,1&2,1&3,1&2&3 valves like we do from the 1st octave, that is considered as the root tone of a brass instrument!) However this doesn't change the fact that the practical range - for all instruments - the 2 octaves and for special artists the 3rd above the second octave (e''' sounding d''') optional final tone of a virtuos solo. A fourth or a fifth valve is enlarging this range and theoretically you can enlarge a) every instrument with by 1 or 2 (and even more valves, if you want). Also you could enlarge the length of the air wave by the trigger or the tuning slide. But it would be wrong and ridiculous to say this is influencing the pitch range of an instrument.)