Are there any free ways to increase the framerate of a video?
what about creating image sequences and then using somehting like ImageMagick to interpolate between the frames.

So (assuming bash, zsh or similar shell) this should work:

mkdir tempdir
mkdir outputdir
ffmpeg -i input.mov -f image2 tempdir/temp_%06d.png #create a png sequence
len=$(ls -1 tempdir|wc -l) #count the files in temp dir
for f in {0..$len} 
do
    old=printf %06d $f #number of current frame
    next=printf %06d $((f+1)) #number of next frame
    new=printf %06d $((f*2)) #number of output frame from current
    newInt=printf %06d $((new+1)) #number of output frame from interpolation
    convert -morph tempdir/temp_${old}.png tempdir/temp_${next}.png outputdir/output_${newInt}.png #create the intermediate frame with imagemagick
    mv tempdir/temp_${old}.png outputdir/output_${new}.png #rename the old frame with a new number and move it to the output dir
done
mv tempdir/temp_${next}.png outputdir/output_${new}.png #move the last frame
ffmpeg -i outputdir/output_%06d.png &lt;&lt;codec etc. options here&gt;&gt; output.mov


basically that creates a temp folder, makes a png image sequence from ffmpeg, creates an intermediate by morphing the two frames into each other using imagemagick, and turns the output image sequence into a movie. 

If this is too slow you could do a quick and dirty interpolation by just blending the two frames together. In that case you'd use 

composite -dissolve 50 tempdir/temp_${old}.png tempdir/temp_${next}.png outputdir/output_${newInt}.png


in place of the convert -morph... line
Obviously you'll need to have imagemagick and ffmpeg installed, and you have to substitute your input and output filenames and preferred output video file format and codec etc options.