Q:

Which particles do not fit into Wigner's picture?

In his accepted and highly upvoted answer to Why particles are thought as irreducible representation in plain English? @Valter Moretti finishes his ADDENDUM with "Finally not all particles fit into Wigner's picture".
Although @Kai subsequently commented "I'm a bit late here but which particles don't fit into Wigner's picture and what is the "enlarged picture" we adopt to accommodate them? – Kai " there doesn't seem to be an answer to this intriguing issue...
May I refresh the question in Kai's comment?

A:

DEFINITIONS AND SUMMARY
A.    (Wigner). A “Particle” is a positive-energy unitary irreducible representation of the Poincare algebra. [1]
B.    Wigner's classification is a classification of the nonnegative (E ≥ 0) energy irreducible unitary representations of the Poincaré group which have sharp mass eigenvalues. [2]
Omitted from this classification are “tachyonic solutions, solutions with no fixed mass, infraparticles with no fixed mass, etc. Such solutions are of physical importance, when considering virtual states”... [2 again]
For the 2nd part of the question: from Kai’s original comment regarding the “enlarged picture” to accommodate the particles not covered by Wigner’s classification: there isn’t one, as the examples showing below should make clear (specifically: it isn’t simply a question of a universal cover or some larger group (beyond a T.O.E.), but rather of varied exceptions to the definition). [UPDATE 19 June 2020: While the Conformal Group contains the Poincaré Group, and so constitutes an “enlarged picture”, as far as I can tell from the following, it still doesn’t account for the exceptions to Wigner's Classification:
physics.stackexchange.com/questions/78660/ “Representations of the Conformal Group in terms of the Poincare Group Reps” physics.stackexchange.com/q/78552/ “Why are there no particles in conformal theories” and physics.stackexchange.com/q/27598/ “Massive excitations in Conformal Quantum Field Theory”.]
Nevertheless, Peter Woit reviews the  Poincare Group and its representations systematically [10], where he highlights in Section 42 the various unphysical representations based on an analysis of orbits in Minkowski space (e.g. tachyons corresponding to the space-like orbits).
DETAIL
Tachyons [3, and refs therein]
“In special relativity, a faster-than-light particle would have space-like four-momentum, in contrast to ordinary particles that have time-like four-momentum. Although in some theories the mass of tachyons is regarded as imaginary, in some modern formulations the mass is considered real, the formulas for the momentum and energy being redefined to this end. Moreover, since tachyons are constrained to the spacelike portion of the energy–momentum graph, they could not slow down to subluminal speeds.
[Wigner’s classification] therefore omits negative-energy states and states with imaginary mass such as tachyonic solutions.”
Infraparticles [4, 5, and refs therein]
“An infraparticle is an electrically charged particle and its surrounding cloud of soft photons—of which there are infinite number, by virtue of the infrared divergence of quantum electrodynamics. That is, it is a dressed particle rather than a bare particle. Whenever electric charges accelerate they emit Bremsstrahlung radiation, whereby an infinite number of the virtual soft photons become real particles. However, only a finite number of these photons are detectable, the remainder falling below the measurement threshold.
The form of the electric field at infinity, which is determined by the velocity of a point charge, defines superselection sectors for the particle's Hilbert space. This is unlike the usual Fock space description, where the Hilbert space includes particle states with different velocities.
Because of their infraparticle properties, charged particles do not have a sharp delta function density of states like an ordinary particle, but instead the density of states rises like an inverse power at the mass of the particle. This collection of states which are very close in mass to m consist of the particle together with low-energy excitation of the electromagnetic field.”
But also:
“The directional charges are different for an electron that has always been at rest and an electron that has always been moving at a certain nonzero velocity (because of the Lorentz transformations). The conclusion is that both electrons lie in different superselection sectors no matter how tiny the velocity is. At first sight, this might appear to be in contradiction with Wigner's classification, which implies that the whole one-particle Hilbert space lies in a single superselection sector, but it is not because m is really the greatest lower bound of a continuous mass spectrum and eigenstates of m only exist in a rigged Hilbert space. The electron, and other particles like it is called an infraparticle.” [My bold]
Presumably, also falling outside Wigner’s classification:

Virtual particles [6]

All of the numerous types of quasiparticle, such as anyons; and collective-excitations, such as phonons.
“These particles are typically called "quasiparticles" if they are related to
fermions, and called "collective excitations" if they are related to bosons, although
the precise distinction is not universally agreed upon. Thus, electrons and electron
holes are typically called "quasiparticles", while phonons and plasmons are typically
called "collective excitations.".“ [7, 8]

Likewise, electrons in topological insulators such as Weyl semimetals [9]

[1] http://pqnelson.github.io/wiki/qft/wigner-classification.html
[2] https://en.wikipedia.org/wiki/Wigner%27s_classification
[3] https://en.wikipedia.org/wiki/Tachyon
[4] https://en.wikipedia.org/wiki/Infraparticle
[5] https://www.researchgate.net/publication/257943101_On_particles_infraparticles_and_the_problem_of_asymptotic_completeness
[6] https://en.wikipedia.org/wiki/Virtual_particle
[7] https://en.wikipedia.org/wiki/Quasiparticle
[8] https://en.wikipedia.org/wiki/List_of_quasiparticles
[9] https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031016-025458
[10] https://www.math.columbia.edu/~woit/QM/fall-course.pdf (see Section 42)

