Starting a non-human PC with more than 1HD - If you start at Level 1 with more than 1d8 hit points (OK, d10 for some classes in 1E) you have effectively got the extra HD on a "mortgage". To pay this "mortgage" off you cannot gain any more hit points until you have completed levels past your starting HD equal to the number of levels for which you were over-rated.
Badly explained. Time for an example.
OK, Bugbear PC. Level 1 but 3HD (ignore pluses unless they are 4+ in which case treat the species as being 1HD more than printed). "Cheating" for Levels 1 and 2, catches up at level 3. To cancel out the bonus gained at Levels 1 and 2 cannot gain hit dice for levels 4 and 5. Starts gaining HD at Level 6 again.
So assuming a Bugbear is just a B/X or BECMI Fighter with surprise ability and 3d8 for hp.
Level 1 - 3d8 hp + CON bonus (CON bonus gained once)
Level 2 - 3d8 hp + CON bonus (CON bonus gained once)
Level 3 - 3d8 hp + CON bonus (CON bonus gained once)
Level 4 - 3d8 hp + CON bonus (CON bonus gained once)
Level 5 - 3d8 hp + CON bonus (CON bonus gained once)
Level 6 - 4d8 hp + CON bonus (CON bonus gained twice)
Level 7 - 5d8 hp + CON bonus (CON bonus gained three times)
and so on.
Neanderthal or Lizardman PC - Both of these species are 2HD creatures, so will be 2HD + Con bonus until reaching Level 4. (Lack of advance at Level 3 "pays off" "cheating" at Level 1).
There is probably a point past which such species unbalance a low level game. You'd have to decide that for yourself. It's also easily abusable by that sort of player so regard it as a litmus test of whether or not a player is the sort of player you'd like to play with.
(Citadel Miniatures ADD56 Bugbear - one of my all-time favourite Citadel castings - halfinched from here - http://www.verdensmand.com/add)