If $\Gamma$ is a compact simply-connected semisimple Lie group, then the Weyl Dimension Formula tells us exactly which dimensions it can act irreducibly on.
For certain $\Gamma$, it is easy to find pairs of nonisomorphic representations of the same dimension:
1). $A_n (n\geq 2)$, $C_n$ ($n$ = A116940(k)), $D_n (n\geq 4)$, and $E_6$ each have pairs of fundamental irreducible representations of the same dimension.
2). Additionally $G_2$ has two irreducible representations of dimension 77.
Furthermore, given that $\Gamma$ has one pair of nonisomorphic representations of the same dimension, it is easy to prove (using the Weyl Dimension Formula) that it has infinitely many such pairs.
Question 1: Among the remaining groups not mentioned above, which are known to have pairs of nonisomorphic irreducible representations of the same dimension?
Question 2: For obvious reasons $A_1$ cannot have such pairs, but are there any other cases where one can rule out the existence of such pairs?
It would seem that since the Dimension Formula so greatly restricts the possible dimensions for the other groups that in the long run there must be pairs of irreducibles of the same dimension; if anyone knows of any results along the lines of such heuristic forcing arguments those would be useful as well.
Edit: Robert's answer below reminded me that such pairs also occur for:
3). $B_n$ when $n$ = A116940(k) just as in the $C_n$ case because of a result I proved awhile back. More generally, one can show that if $B_n$ has such a pair then so does $C_n$ and conversely if $C_n$ has such a pair then so does $B_n$, so Robert's comment also shows that $C_4$ and $C_5$ have such pairs (the pairs arising from Robert's examples are in dimensions 11354112 and 38928384 for $C_4$ and 24741150720 for $C_5$).
4). $F_4$ has two irreducible representations of dimension 1053 that I had completely forgotten about in my list.