How many electrons are there – quantum-wise? If you consider that a particle exists as a quantum field, could you say that all the particles' fields combine into one field for that particle type? Why could you then not say there is no specific number of particles represented, and that they only manifest themselves when being observed?
This concept came up when there was a discussion of the Higgs boson, and one person said there was only one Higgs boson and it created its field across the universe.
Could you say that there is only one electron in the universe, but it is being observed, in some sense, fairly often?
 A: The "number of electrons" is normally an extremely useful and rather well-defined quantity and the number of electrons in the Universe is surely much greater than one. Each atom with $Z$ protons in the nucleus contributes $Z$ electrons to the total number of electrons. Then add the free electrons, and so on, you will get a large number.
In quantum field theory, the number of electrons is a linear operator: in any quantum mechanical theory, every quantity that is observable by one measurement is represented by a linear Hermitian operator. The only extra subtleties is the existence of antiparticles of the electrons, the positrons. There exists not only $N_{e^-}$, the number of "electrons proper", but also the total charge carried by the electrons and positrons,
$$Q_e = -N_{e^-} + N_{e^+}$$
If there were (almost) as many positrons in the Universe as the number of electrons, the two terms would (almost) cancel, producing a very small $Q_e$. It often makes sense to count the positron as "minus one electron". Even more precisely, a positron may be viewed as an "electron moving backwards in time". According to Wheeler, you could have one electron going back and forth in time through spacetime, and it would manifest itself as all electrons and positrons in the world.
This is mostly a popular physics meme only. One problem is that the number of positrons that are left in the Universe is by far smaller than the number of electrons. Most antimatter has annihilated away. The total electric charge of the Universe is (basically) zero but one would have to include the particles of all other species to make Wheeler's idea viable.
To summarize, there is no useful sense in which the number of electrons only has to be zero or one. The same thing holds for the Higgs bosons, however. There is no useful sense in which the number of Higgs bosons is zero or one. When it comes to the Higgs field filling the space, it has a "vacuum expectation value" which may be viewed as a "condensate of many Higgs bosons". How many Higgs bosons the Universe has depends on various details of the definition. It may be large or "zero" in the vacuum. But even if one adopts the "common" definition of $N_h$ that has $N_h=0$ in the vacuum, it's still possible to increase $N_h$ by producing new Higgs bosons, e.g. at the LHC. They decay quickly but across the Universe, there are certainly many Higgs bosons at one moment.
I suspect that the claim that there is only "one Higgs boson" meant that so far, we have only discovered one "type" or "species" of the Higgs boson, the Higgs boson of the mass 125 GeV. But it has many "exact clones" in the Universe.
A meta-comment: I think that the question by the OP may be classified as a question of the type "I invented an amusing vague meme that sounds a little bit like physics, and I ask others to show their excitement about it". But physics doesn't quite work like that. Physics tries to achieve a certain precision and quantitative level of predictions. Sometimes, the path towards complete theories goes through vague ideas and feelings. But this is a server meant to answer questions and physics questions may only be answered when we deal with sufficiently well-defined theories and questions – pretty much inevitably with theories forming the established body of knowledge. So as a piece of physics art, the question could be said to be interesting, playful, B-graded, or something like that. But genuine physics questions talk about propositions that are right or wrong (and all the terms are well-defined and guaranteed to be measurable and meaningful by the person who asks) and the statements in the question above aren't really of that type (because the question deliberately wants to use e.g. a non-standard and ill-defined meaning of the term "the number of electrons"). So I would recommend the OP to try to learn how those things work at a much sharp level of precision.
A: 
If you consider that a particle exists as a quantum field, could you say that all the particles' fields combine into one field for that particle type?

This is essentially correct, but it's wrong on the details. A particle is not a quantum field. A quantum field is a big vibratey thing that can be either in a ground state or one of many excited states. Like all quantum theories, you can generally have lots of such excitations at once and such excitations "come in lumps", called quanta. Particles are the quanta of excitation in the quantum field. So there is in our modern understanding one electron-field with many excitations, called electrons. 
It is slightly more nuanced even than that, because positrons (anti-electrons) are also 
"excitations" of this field in a manner of speaking. Essentially the quantum field is in a "half-full state": it has a spectrum of energies which does not just go to $+\infty$ but also to $-\infty,$ and is full with infinitely many "electrons" in just such a way that it all fills up to $0$ energy in the "vacuum" case. If we "excite" one of these quanta from some energy $E_0 < 0$ to some energy $E_1 > 0$ by adding the energy $E_1 - E_0$ to the system, then we have performed a "pair production": the new "electron" (quanta occupying the normally-unoccupied positive-energy state) is balanced out by this "positron" (missing-quanta or "hole" sitting in a normally-occupied negative-energy state). All of the forces-on-electrons will act to move the rest of the quanta "around" this hole, and so the hole will act just like an electron, but with opposite charge. 
If we call all of the electrons that are filling up the energies $E < 0$ as "invisible" electrons, then positrons are holes in the space of invisible electrons, and pair production consists of making an electron "visible" by leaving a hole in its wake.
If you are wondering "why doesn't another "invisible" electron just fall into the hole?" the answer is that yes this can happen to a limited extent, but it looks like normal everyday "collisions" that happen to an electron as well, it does not suffice to "bubble up" the hole to $E=0$. The essential problem is that there is a band gap created by the finite rest mass of the electron, so that eventually a positron at rest will be at the same energy as all of the invisible electrons that would potentially fill it. Then it can only really be destroyed by absorbing a visible electron.

Why could you then not say there is no specific number of particles represented, and that they only manifest themselves when being observed?

I mean, you can. It is possible for us to arrange a Schrodinger's cat experiment where instead of killing a cat we fire some positrons that we have in a circular orbit into a block of lead. In this case the number of "visible" electrons will certainly decrease in one case but not in the other, and we'll have to view the system as being in a quantum superposition of different numbers-of-particles states. 
(The mathematics that we use to keep ourselves sane when we do this is called Fock space, and it represents one of the first places you can take quantum mechanics where Schrodinger's wavefunction is no longer helpful. In turn, the idea to look at a quantum field can in some ways come from looking at the Fock space construction and saying, "how do I get a number-density akin to my old $|\psi|^2$ out of this thing?!" This analogy isn't complete because full QFT involves things like Lagrangians, but it is good enough for a lot of condensed-matter theory to be done this way.)
However, this method of speaking breaks down a bit if you count the "invisible" electrons as well, so that the positron has "electron number -1" and we did not actually violate any count of particles. Okay: so then we might consider muons, which are "heavy electrons" that can turn into true electrons through a process called the "weak interaction." But maybe you want to include those muons as "electrons" too.  Fine, then we have beta-minus decay: a neutron turns into a proton by emitting an electron and an electron antineutrino. But maybe you want to include those antineutrinos as "holes" in the neutrino space which are equivalent to electrons...
The formal name for what you are counting here, L = electrons + muons + taus + neutrinos - antiparticles, is called the "total lepton number". As far as we know, it is conserved by all interactions, and the total number of leptons in the universe is 100% constant. (If it is not conserved, then at least it is thought that a quantity called variously "weak hypercharge" or $B - L$ is conserved: many grand-unified theories allow a proton to decay into a positron and a pion; this would mean that we can still do our thing if we count protons as if they were antiparticles too.)
A: Let me attempt a pop science version of Luboš' answer:


*

*A quantum feld is an operator valued field that occupies all of spacetime. So there is just a single electron field (though be cautious about thinking of this field as a physical object because it isn't).

*A quantum field can be excited and gain energy, or it can relax and lose energy. In both cases the energy is gained or lost in quanta, and those quanta are what we call particles.
So if you start in the vacuum state, with no particles present, then transfer a quantum of energy to the electron field an electron will appear. Transfer more energy and a second electron appears and so on. For example this is how particles are created in a collider - the kinetic energy of the colliding particles can be transferred into a quantum field where it creates a new particle. This approach also neatly explains why all electrons are identical. It's because they are all excitations of the same quantum field.
To get back to your question, no, it is not correct to say there is only one electron or only one Higgs or of any other particle. However it is correct to say there is only one electron field, only one Higgs field and so on. Arbitrarily large numbers of electrons, Higgs, or whatever can be created by transferring energy into the corresponding quantum fields subject of course to the energy being available and the various conservation laws being observed.
