This question was just raised by a colleague (who shall for the moment remain anonymous). It may or may not have a reasonable answer. 



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> For which finite nonabelian groups $G$ do all irreducible complex characters of degree $>1$ have at most one strictly positive real value (namely the degree)?
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One small example is the alternating group $A\_4$. This could be viewed as a sort of degenerate Chevalley group, whose Steinberg character of degree $3$ has other values
$-1, 0, 0$. But for most finite groups of Lie type over a finite field of order $q$, the Steinberg character of degree a power of $q$ will take on more than one positive integral value. While an enumeration of finite groups having the stated property may well be out of reach, it would be of interest to know:



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> Are there infinitely many groups $G$ with the special property stated above?
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Finite nonabelian groups with few positive real character values?
Here are some classes of such groups I can think of:


* extraspecial $p$-groups (and probably some other $p$-groups)
* metacyclic groups, where $\lvert G: G'\rvert$ has odd order
* the groups $AGL(1,q)$ (this subsumes Tim Dokchitser's examples and the $A\_4$ example). More generally, suppose a group $H$ with few positive real values acts sharply transitive on the nonzero elements of a finite vector space $V$. Then the semidirect product $HV$ has also only characters with at most one strictly positive value. An example is $Q\_8$ acting on $C\_3\times C\_3$.


Here is an argument that rules out some groups: If a group $G$ acts $2$-transitively on some set, then the permutation character $\pi$ has the form $\pi= 1+\chi$ with $\chi \in \operatorname{Irr}(G)$. By assumption, if $g$ acts nontrivially, then $\chi(g)\leq 0$. This implies that every element not in the kernel $K$ of the action has at most one fixpoint. But then $G/K$ is a Frobenius group, the elements with no fixpoints form the Frobenius kernel.  

While I'm not an expert for finite simple groups, I suppose this argument rules out many (all?) finite simple groups, namely all that admit a $2$-transitive permutation representation.  

Since the property in the question is inherited by factor groups, this would also yield that there are no perfect groups with that property.  

EDIT: Thanks to Jack for putting that right. Anyway, simple groups are ruled out by ARupinksi's comment.


