Through years of working with regiments, orders of battles, etc., I managed to accidentally memorize the basic regimental histories of every ACW unit. Many upsides to this, but also many downsides -- the greatest of these being that no, not every regiment is very interesting. For instance:
- The 5th, 6th, 7th, and 9th Delaware never saw combat
- The 67th-71st, 132nd-136th, 138th, 139th, 141st, 144th, and 145th-156th Illinois never saw combat
- The 102nd-114th Indiana and most of the 130, 140, and 150s never saw combat
- The 41st-48th Iowa never saw combat
- The 30th Michigan, as @Kurtlg mentioned, never saw combat
- The 11th Minnesota never saw combat
- Over half of all Missouri regiments (infantry, cavalry, and artillery) never left the state and/or saw combat
- I believe only seven of the forty-one Ohio National Guard units saw combat, and most of the 180s and 190s never saw combat
- Most of Wisconsin's 40-series regiments never saw combat
However, what I've seen is that this only goes for Union units -- every Confederate unit raised saw combat or special duty in some shape or form. Every Confederate regiment deserves a history, but not all Union units deserve a history; I'd limit it to those that saw combat.