As a FYI, it's 26 XP to get a level 6 unit (unless you are Charismatic, in which case it's 20).
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The number of Promotions you need for a unit is deceiving for Aggressive and Protective leaders, Just focus on the number of XP you get.
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level 6 needed for West Point is in warlords, so I assume you play warlords.
For WP to be available you need to have a unit promoted 5 times through XPs.
Level of Xps required for the various levels :
Level 1 : 0xp.
Level 2: 2xp
level 3 : 5 (4 if charismatic)
level 4 : 10 (8 if char)
level 5 : 17 (13 if char)
level 6 : 26 (20 if char)
(level 7 is at 37 [28], level 8 is at 50 [38])
Where do you get xps?
- from barracks, stables and civics (theocracy, vassalage) in the city where you build the unit. Note that military instructors (=settled GG) also give XPs to units built in this city.
- from combat. Each combat gives you 1 to 10 xps, depending on if you attack or defend and what the relative strength is. SO winning at 5% odds will give you 10 xps, while mopping up at 99,9% odds will give you 1 xp.
- from attached Great Generals. A warlord gives 20 xps divided amongst the units in the stack. for example, you have a spear and an axe in a stack. The spear has 3 xps, the axe 7. If you attach the GG to the axe, the spear will be at 13 xps and the axe at 17.
How difficult is it to get to 26?
It's a real challenge without a GG. Most of your fights will give you 1 or 2 xps. those giving more than this are long shots that you'd better not even try

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Let's start with a chariot built in a city with a stable (unlikely, but who knows), running theocracy.
You start with 7 xps.
It would require 10 "not quite sure, but likely winnable" (= above 80% odds but less than 98%) fights to get to 26.
Or 19 retreats (yes you get 1 xp for retreating)/sure fights. That's a lot.
Note that fighting animals can't bring you over 5 xp and fighting barbs won't bring you above 10 xps.
That's why so many people use the first GG to attach to a single unit above 6 xps, getting a level 6 unit straight away.