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.

fig1