The Hopf fibration is too mainstream. Instead of covering the 3-sphere with smooth curves, let’s cover it with surfaces. 

The Reeb foliation of the 3-sphere has a relatively simple explanation:
- Embedding a central torus C in the 3-sphere partitions it into two solid tori
- Within each solid torus, the leaves are all identical, just rotated copies of each other. Each can be interpreted as the surface of revolution of a curve that starts with a vertical tangent then tends to a horizontal asymptote, but wrapped around itself inside the torus (see below):

c=0.4; (* determines the ratio of the torus' tube radius to full radius *)
F[x_]:=(Sqrt[(2/π)ArcTan[x]]); (* The leaf's shape function *)
TorusPoints[θ_,α_,f_]:={Cos[θ],Sin[θ],0}+c f[θ] (Cos[α] {Cos[θ],Sin[θ],0}+Sin[α] {0,0,1});(* Plotting points around the torus but with varying depth parameter f depending on θ *)

Plot[{F[x],1},{x,0,5},PlotRange->All] 
ParametricPlot3D[{x,Sin[t]F[x],Cos[t]F[x]},{x,0,5},{t,0,2π},Mesh->None,MaxRecursion->5]
ParametricPlot[{TorusPoints[θ,0,F][[1;;2]],{Cos[θ],Sin[θ]},(1+c){Cos[θ],Sin[θ]}},{θ,0,2.5π},Mesh->None]
ParametricPlot3D[TorusPoints[θ,α,F],{θ,0,2.5π},{α,0,2π},PlotStyle->Opacity[0.5],Mesh->None,MaxRecursion->5]

To visualize the Reeb foliation, we will use the stereographic projection of the 3-sphere onto (ℝ^3). We will set up a standard torus C and foliate the inside as above.

insides=Table[ParametricPlot3D[TorusPoints[θ,α,F[#+k]&],{θ,-k,π-k},{α,-π,0},Mesh->None,PlotStyle->{Hue[k/30],Specularity[White,100]},MaxRecursion->5],{k,0,2π-π/6,π/6}];
sampleinside=ParametricPlot3D[TorusPoints[θ,α,F],{θ,0,4π/3},{α,-π,π},Mesh->None,PlotStyle->{Hue[0],Specularity[White,100]},MaxRecursion->5];
Ctorus=ParametricPlot3D[TorusPoints[θ,α,1&],{θ,0,2π},{α,-π-.4,.4},Mesh->None,PlotStyle->{Blue,Specularity[White,100]},MaxRecursion->5];

Show[{sampleinside,insides,Ctorus},PlotRange->All,Boxed->False,Axes->False]

Foliating the outer solid torus takes a bit more thinking. We had a nice parametrization of the inside of C; now we need one for the exterior. One parameter will simply be angle around the z-axis, but then we have to parametrize a half-plane-minus-a-circle (the cross-section of the exterior of C at a given angle around the z-axis). The idea is that we can use a Mobius transformation to parametrize this space by comparing it to a suitable annulus. 

The function Mobius takes the unit circle to the vertical axis:

Mobius[{x_,y_}]:={Re[(((x+I y)-1)/((x+I y)+1))],Im[(((x+I y)-1)/((x+I y)+1))]};

We can identify the annulus we need by finding the preimage of three points on the circle to be removed from the half-plane; it is straightforward to see that the circle centered at 1 with radius c comes from the circle centered at -1 with radius 2/c. Thus, we parametrize the annulus between this circle and the unit circle (the level sets must meet the unit circle at right angles), and push the parametrization forward using the Mobius function.

AnnulusParam[β_,t_]:=(maxt=-Cos[β]+(Sqrt[((((2/c)))^2)-((Sin[β])^2)]);(1+t (maxt-1)){Cos[β],Sin[β]})

Illustrating the parametrization of the annulus, and how it transforms into a parametrization of the half-plane-minus-disk. Also included are the desired cross-sections of leaves. Note that they must meet the z-axis at right angles in order for their surface of revolution to be smooth.

coordEx=Table[{AnnulusParam[#1,#2],0,2π,{LightGray}}&[xx,t],{t,0,1,1/6}] ~Join~Table[{AnnulusParam[#1,#2],0,1,{LightGray}}&[β,xx],{β,0,2π,π/8}]~Join~
Table[{AnnulusParam[#1,F[#1+#2]],If[#2==π,-1.01π,-#2],
2π-#2,{Hue[#2/(2π)]}}&[xx,k],{k,0,2π-π/4,π/4}];
Show[ParametricPlot[#[[1]]/.xx->x,{x,#[[2]],#[[3]]},PlotStyle->#[[4]]]&/@coordEx,PlotRange->All]
Show[ParametricPlot[Mobius[#[[1]]]/.xx->x,{x,#[[2]],#[[3]]},PlotStyle->#[[4]]]&/@coordEx,PlotRange->{{0,1+2c},{-3c,3c}}]

We can finally define a function OuterTorusPoints that will allow us to plot the outer leaves, and then we simply need to put all the pieces together!

OuterTorusPoints[θ_,β_,f_]:={Cos[θ]Mobius[AnnulusParam[β,f[β]]] [[1]],Sin[θ]Mobius[AnnulusParam[β,f[β]]] [[1]],Mobius[AnnulusParam[β,f[β]]] [[2]]};

qual=8; (* 2 for quick results. Up to 8 for very high resolution *)

leaves={-0.93π,-0.8π,-0.65π,-0.45π,0,0.55π,0.7π,0.84π,0.93π,π};
outers=ParametricPlot3D[OuterTorusPoints[θ,β,Function[x,F[x+#]]],{θ,0,π},{β,-#,2π},
Mesh->None,PlotStyle->{Hue[#/(10π)+1/3],Specularity[White,100]},MaxRecursion->qual]&/@leaves;

insides2=Table[ParametricPlot3D[TorusPoints[θ,α,F[#+k]&],{θ,-k,π-k},{α,-π-.4,-.4},Mesh->None,PlotStyle->{Hue[k/30],Specularity[White,100]},MaxRecursion->qual],{k,0,2π-π/6,π/6}];

sampleinside2=ParametricPlot3D[TorusPoints[θ,α,F[#+π/6]&],{θ,-π/6,4π/3},{α,-π,π},Mesh->None,PlotStyle->{Hue[Pi/90],Specularity[White,100]},MaxRecursion->qual];
Ctorus2=ParametricPlot3D[TorusPoints[θ,α,1&],{θ,0,2π},{α,-π+.2,-.2},Mesh->None,PlotStyle->{Blue,Specularity[White,100]},MaxRecursion->qual];

I could use a new desktop background...  (“Rasterize” added by DBN to decrease file size)

Show[{insides2,outers,sampleinside2,Ctorus2},PlotRange->{{-8,4},{-4,8},{-4,4}},Boxed->False,Axes->False,Background->Gray,ImageSize->{1296,768},ViewVector->{{10,-20,10},{-0.9,0,0}},ViewAngle->π/24] // Rasterize