Think of it. An entire playthrough all the way to level 20 with the restriction that your chosen worst SPECIAL attribute must stay at it's original base of 1. You cannot use intense training to raise it. Nor can you take the bobblehead to raise it. However, you can raise it through worn clothing items or chems, and even chems from Better Living Through Chems. What SPECIAL attribute would challenge your game the most under these rules?

For me it would be luck. I need to stagger hostiles because my low endurance demands that I avoid being shot, and by staggering them ( supermutants and overlords ) I can prevent them from firing while whittling them down. With fDamageWeaponsMult at 5.4 I can't make a real stand against MMMF3 spawns at 2-4. Unless I have a river of stimpaks hotkeyed.

Strength, mostly. Also Perception.

Intelligence or Luck.

Here's my view of the effects of each at 1:

Stength: You can carry less, but still 160 should be more than enough to carry everything you need plus some loot. Do less melee damage, but there are plenty of combat alternatives, and a sneak attack with a melee weapon will still do decent damage.

Perception: You don't see enemies on your compass as quickly. Some people play with the compass off completely, this will have little impact on your game.

Endurance: You only gain 20 extra HP from End, and this can be countered with the use of additional stim packs. Poison and radiation resistance are not particularly important.

Charisma: Most of my characters start with a Cha of 1, so I would not need to do anything unusual to counter it. Game impact is minimal.

Intelligence: The worst of the lot to have at 1, IMO. With an Int of 1 you can't take the Comprehension or Educated perks, which means 11 skill points per level and 1 SP per skill book. This would force you to sacrifice some skills in order to have decent scores in others. Since most of the actual game depends on skill usage, this seems to me like the biggest handicap.

Agility: You have fewer AP. This can be largely countered by the use of chems, though it does preclude the Action Boy/Girl perks as they require an Agility of 6.

Luck: You do a lot fewer critical hits, and you can't take Better Criticals. If you rely on criticals a lot then this could hurt you. On the other hand, you can simply play a sneaky character and get lots of sneak attack criticals anyway, so there are ways to play around this.

In summation, Intelligence is the only SPECIAL that has a significant handicap associated with a low score that you cannot easily work around. But in all honesty even this handicap is not hard to work around, it just means you have to choose some skills to sacrifice in favor of the more important ones. SPECIALs are not all that important in this game, the main impact of putting them at 1 is to limit your choice of perks.