A few additional work-arounds:

1. Post-process Plot output to remove extra points: Use

Plot[...] /. Point -> Point @* First 

in the first argument of Manipulate to get:

2. Use the option

 DisplayFunction -> (# /. Point -> Point@*First &) 

to get the same result.

3. Use a function for the value of the option MeshStyle and use the replacement rule above in that function:

MeshStyle -> ({Directive[Black, PointSize[0.02]], # /. Point -> Point@*First} &) 

same picture

4. Remove b and b/2 from the function list in the first argument of Plot and show them using GridLines:

Manipulate[Plot[n[n0, b, t, r], {t, 0, 20}, PlotStyle -> Directive[Thick, Pink], MeshFunctions -> {#2 &}, Mesh -> {{b/2}}, MeshStyle -> Directive[Black, PointSize[0.02]], GridLines -> {None, {{b, Directive[Opacity[1], Purple, Thick]}, {b/2, Directive[Blue, Dashed, Thin]}}}], {{n0, 10}, 1, 100, 1, Appearance -> "Labeled"}, {{b, 100}, 0, 100, 5, Appearance -> "Labeled"}, {{r, 0.4}, 0, 5, 0.1, Appearance -> "Labeled"}] 

A few additional work-arounds:

1. Post-process Plot output to remove extra points: Use

Plot[...] /. Point -> Point @* First 

in the first argument of Manipulate to get:

2. Instead of post-processing outside Plot use the option

 DisplayFunction -> (# /. Point -> Point@*First &) 

inside Plot to get the same result.

3. Use a function for the value of the option MeshStyle and use the replacement rule above in that function:

MeshStyle -> ({Directive[Black, PointSize[0.02]], # /. Point -> Point@*First} &) 

same picture

4. Remove b and b/2 from the function list in the first argument of Plot and show them using GridLines:

Manipulate[Plot[n[n0, b, t, r], {t, 0, 20}, PlotStyle -> Directive[Thick, Pink], MeshFunctions -> {#2 &}, Mesh -> {{b/2}}, MeshStyle -> Directive[Black, PointSize[0.02]], GridLines -> {None, {{b, Directive[Opacity[1], Purple, Thick]}, {b/2, Directive[Blue, Dashed, Thin]}}}], {{n0, 10}, 1, 100, 1, Appearance -> "Labeled"}, {{b, 100}, 0, 100, 5, Appearance -> "Labeled"}, {{r, 0.4}, 0, 5, 0.1, Appearance -> "Labeled"}] 

A few additional work-arounds:

1. Post-process Plot output to remove extra points: Use

Plot[...] /. Point[x_] :> Point[First@x] 

in the first argument of Manipulate to get:

2. Remove b and b/2 from the function list in the first argument of Plot and show them using GridLines:

Manipulate[Plot[n[n0, b, t, r], {t, 0, 20}, PlotStyle -> Directive[Thick, Pink], MeshFunctions -> {#2 &}, Mesh -> {{b/2}}, MeshStyle -> Directive[Black, PointSize[0.02]], GridLines -> {None, {{b, Directive[Opacity[1], Purple, Thick]}, {b/2, Directive[Blue, Dashed, Thin]}}}], {{n0, 10}, 1, 100, 1, Appearance -> "Labeled"}, {{b, 100}, 0, 100, 5, Appearance -> "Labeled"}, {{r, 0.4}, 0, 5, 0.1, Appearance -> "Labeled"}] 

A few additional work-arounds:

1. Post-process Plot output to remove extra points: Use

Plot[...] /. Point -> Point @* First 

in the first argument of Manipulate to get:

2. Use the option

 DisplayFunction -> (# /. Point -> Point@*First &) 

to get the same result.

3. Use a function for the value of the option MeshStyle and use the replacement rule above in that function:

MeshStyle -> ({Directive[Black, PointSize[0.02]], # /. Point -> Point@*First} &) 

same picture

4. Remove b and b/2 from the function list in the first argument of Plot and show them using GridLines:

Manipulate[Plot[n[n0, b, t, r], {t, 0, 20}, PlotStyle -> Directive[Thick, Pink], MeshFunctions -> {#2 &}, Mesh -> {{b/2}}, MeshStyle -> Directive[Black, PointSize[0.02]], GridLines -> {None, {{b, Directive[Opacity[1], Purple, Thick]}, {b/2, Directive[Blue, Dashed, Thin]}}}], {{n0, 10}, 1, 100, 1, Appearance -> "Labeled"}, {{b, 100}, 0, 100, 5, Appearance -> "Labeled"}, {{r, 0.4}, 0, 5, 0.1, Appearance -> "Labeled"}]