Here is a list of races and what classes they can be. In brackets of the first mention of the class and what the maximum level can be. U = unlimited

Dwarf
F (9)
T (U)
F / T

Elf
F (7)
M (11)
T (U)
F / M
F / T
F / M / T
M / T

Gnome
F (6)
T (U)
F / T


Halfling
F (6)
T (U)
F / T

Half Elf
C (5)
F (8)
M (8)
T (U)
C / F
C / F / M
C / M
F / M
F / T
F / M / T
M / T

Human
C (U)
F (U)
M (U)
T (U)

Since halflings can't have more than 17 str in the game they can't have more than 5 fighter levels either.