plotting – Find exponential function that becomes constant at 1

samplelist={{0., 15}, {0.031746, 14}, {0.0634921, 13}, {0.0952381, 
  12}, {0.126984, 11}, {0.15873, 11}, {0.190476, 10}, {0.222222, 
  9}, {0.253968, 9}, {0.285714, 8}, {0.31746, 8}, {0.349206, 
  7}, {0.380952, 7}, {0.412698, 7}, {0.444444, 6}, {0.47619, 
  6}, {0.507937, 6}, {0.539683, 5}, {0.571429, 5}, {0.603175, 
  5}, {0.634921, 4}, {0.666667, 4}, {0.698413, 4}, {0.730159, 
  4}, {0.761905, 4}, {0.793651, 3}, {0.825397, 3}, {0.857143, 
  3}, {0.888889, 3}, {0.920635, 3}, {0.952381, 3}, {0.984127, 
  2}, {1.01587, 2}, {1.04762, 2}, {1.07937, 2}, {1.11111, 
  2}, {1.14286, 2}, {1.1746, 2}, {1.20635, 2}, {1.2381, 2}, {1.26984, 
  2}, {1.30159, 2}, {1.33333, 1}, {1.36508, 1}, {1.39683, 
  1}, {1.42857, 1}, {1.46032, 1}, {1.49206, 1}, {1.52381, 
  1}, {1.55556, 1}, {1.5873, 1}, {1.61905, 1}, {1.65079, 1}, {1.68254,
   1}, {1.71429, 1}, {1.74603, 1}, {1.77778, 1}, {1.80952, 
  1}, {1.84127, 1}, {1.87302, 1}, {1.90476, 1}, {1.93651, 
  1}, {1.96825, 1}, {2., 1}}

Show(ListPlot(samplelist), 
 Plot(Exp(-t /(0.5)) samplelist((1, 2)), {t, 0, 2}))

fig2


I need an exponentially decaying function that fits my data and becomes constant (equal to 1 here) ultimately.

fig3