In the modular case (i.e., when the characteristic of the field divides the order of the group), it is more interesting to look for indecomposable representations (because there are very few irreps or, as they are known in this context, simple modules: the 'worst' is that of a $p$-group, which has one simple module!), and those can be as big as you want, in general, as soon as the group is not of finite representation type. This follows from the first Brauer-Thrall conjecture, proved by Maurice Auslander et al.; see, for example, [Ringel, Claus Michael. On algorithms for solving vector space problems. I. Report on the Brauer-Thrall conjectures: Rojter's theorem and the theorem of Nazarova and Rojter. Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), pp. 104--136, Lecture Notes in Math., 831, Springer, Berlin, 1980. MR0607142] and the references therein.

The smallest non-trivial example is the Klein Klein four group $\mathbb Z_2\oplus\mathbb Z_2$ in characteristic two, which is of infinite tame representation type, so has indecomposable modules of arbitrary high dimension. They have been known for ages, in various forms; they are described nicely, e.g., in [Benson, D. J. Representations and cohomology. I. Basic representation theory of finite groups and associative algebras. Second edition. Cambridge Studies in Advanced Mathematics, 30. Cambridge University Press, Cambridge, 1998. xii+246 pp. MR1644252].

In the modular case (i.e., when the characteristic of the field divides the order of the group), it is more interesting to look for indecomposable representations (because there are very few irreps or, as they are known in this context, simple modules: the 'worst' case is that of a $p$-group, which has one simple module!), and those can be as big as you want, in general, as soon as the group is not of finite representation type. This follows from the first Brauer-Thrall conjecture, proved by Maurice Auslander et al.; see, for example, [Ringel, Claus Michael. On algorithms for solving vector space problems. I. Report on the Brauer-Thrall conjectures: Rojter's theorem and the theorem of Nazarova and Rojter. Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), pp. 104--136, Lecture Notes in Math., 831, Springer, Berlin, 1980. MR0607142] and the references therein.

The smallest non-trivial example is the Klein Klein four group $\mathbb Z_2\oplus\mathbb Z_2$ in characteristic two, which is of infinite tame representation type, so has indecomposable modules of arbitrary high dimension. They have been known for ages, in various forms; they are described nicely, e.g., in [Benson, D. J. Representations and cohomology. I. Basic representation theory of finite groups and associative algebras. Second edition. Cambridge Studies in Advanced Mathematics, 30. Cambridge University Press, Cambridge, 1998. xii+246 pp. MR1644252].

In the modular case (i.e., when the characteristic of the field divides the order of the group), it is more interesting to look for indecomposable representations (because there are very few irreps or, as they are known in this context, simple modules: the 'worst' is that of a $p$-group, which has one simple module!), and those can be as big as you want, in general, as soon as the group is not of finite representation type. This follows from the first Brauer-Thrall conjecture, proved by Maurice Auslander et al.; see, for example, [Ringel, Claus Michael. On algorithms for solving vector space problems. I. Report on the Brauer-Thrall conjectures: Rojter's theorem and the theorem of Nazarova and Rojter. Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), pp. 104--136, Lecture Notes in Math., 831, Springer, Berlin, 1980. MR0607142] and the references therein.

The smallest trivial example is the Klein Klein four group $\mathbb Z_2\oplus\mathbb Z_2$ in characteristic two, which is of infinite tame representation type, so has indecomposable modules of arbitrary high dimension. They have been known for ages, in various forms; they are described nicely, e.g., in [Benson, D. J. Representations and cohomology. I. Basic representation theory of finite groups and associative algebras. Second edition. Cambridge Studies in Advanced Mathematics, 30. Cambridge University Press, Cambridge, 1998. xii+246 pp. MR1644252].

In the modular case (i.e., when the characteristic of the field divides the order of the group), it is more interesting to look for indecomposable representations (because there are very few irreps or, as they are known in this context, simple modules: the 'worst' is that of a $p$-group, which has one simple module!), and those can be as big as you want, in general, as soon as the group is not of finite representation type. This follows from the first Brauer-Thrall conjecture, proved by Maurice Auslander et al.; see, for example, [Ringel, Claus Michael. On algorithms for solving vector space problems. I. Report on the Brauer-Thrall conjectures: Rojter's theorem and the theorem of Nazarova and Rojter. Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), pp. 104--136, Lecture Notes in Math., 831, Springer, Berlin, 1980. MR0607142] and the references therein.

The smallest non-trivial example is the Klein Klein four group $\mathbb Z_2\oplus\mathbb Z_2$ in characteristic two, which is of infinite tame representation type, so has indecomposable modules of arbitrary high dimension. They have been known for ages, in various forms; they are described nicely, e.g., in [Benson, D. J. Representations and cohomology. I. Basic representation theory of finite groups and associative algebras. Second edition. Cambridge Studies in Advanced Mathematics, 30. Cambridge University Press, Cambridge, 1998. xii+246 pp. MR1644252].