I voted to close this as a duplicate but since the OP said "it seems too complicated for what I needed to do" here is a way to do it.

Copy the whole code given by Jens and modify the function legendMaker by commenting Line[...] in f:

If[#1 === {None} || (PlotStyle /. {opts}) === None, {}, Line[{{-.1,0}, {.1,0}}]]

should be:

If[#1 === {None} || (PlotStyle /. {opts}) === None, {}, (* Line[{{-.1,0}, {.1,0}} *)]]

_{You can simply remove the whole line but commenting it allows you to use it afterwards.}

Then simply use:

function[gh_, ga_] := If[gh == ga, 1 + 0.18^((gh + ga)^0.25), 1 + 0.18^((gh - ga)^2)^0.25];
p = DiscretePlot[{function[0, i]^i, function[0 + 1, i]^i, function[0 + 2, i]^i}, {i, 0, 30},
  PlotRange -> Full];

opts = Sequence[Background -> Transparent, RoundingRadius -> 10];
Overlay[{p, 
  legendMaker[{"1", "2", "3"}, 
   PlotStyle -> (ColorData[1][#] & /@ Range@3), 
   PlotMarkers -> {\[FilledSmallCircle]}, opts]}, 
 Alignment -> {Right, Top}]

I voted to close this as a duplicate but since the OP said "it seems too complicated for what I needed to do" here is a way to do it.

Copy the whole code given by Jens and modify the function legendMaker by commenting Line[...] in f:

If[#1 === {None} || (PlotStyle /. {opts}) === None, {}, Line[{{-.1,0}, {.1,0}}]]

should be:

If[#1 === {None} || (PlotStyle /. {opts}) === None, {}, (* Line[{{-.1,0}, {.1,0}} *)]]

_{You can simply remove the whole line but commenting it allows you to use it afterwards.}

Then simply use:

function[gh_, ga_] := If[gh == ga, 1 + 0.18^((gh + ga)^0.25), 1 + 0.18^((gh - ga)^2)^0.25];
p = DiscretePlot[{function[0, i]^i, function[0 + 1, i]^i, function[0 + 2, i]^i}, {i, 0, 30},
  PlotRange -> Full];

opts = Sequence[Background -> Transparent, RoundingRadius -> 10];
Overlay[{p, 
  legendMaker[{"1", "2", "3"}, 
   PlotStyle -> (ColorData[1][#] & /@ Range@3), 
   PlotMarkers -> {\[FilledSmallCircle]}, opts]}, 
 Alignment -> {Right, Top}]