The following code computes lower bounds for the smallest possible such $c$. After 15 minutes of computation, my laptop got that $c>40$ (improveable by more computation time).
Here is how the code works. Assume that such $c$ exists. We know that $\lfloor c\rfloor$ is prime, so $c \ge 2$. Then $c^2 \ge 4$, but $\lfloor c^2 \rfloor$ is prime, so $c^2 \ge 5$ and then $c \ge 5^{1/2}$. Now, $\lfloor 5^{1/2} \rfloor = 2$, $\lfloor 5^{2/2} \rfloor = 5$, $\lfloor 5^{3/2} \rfloor = 11$, $\lfloor 5^{4/2} \rfloor = 25$. But $c \ge 5^{1/2}$, so $\lfloor c^4 \rfloor \ge \lfloor 5^{4/2} \rfloor = 25$ and $\lfloor c^4 \rfloor$ is prime, so $\lfloor c^4 \rfloor \ge 29$ (the next prime after $25$). Thus $c \ge 29^{1/4}$. Now $\lfloor 29^{1/4} \rfloor = 2$, $\lfloor 29^{2/4} \rfloor = 5$, $\lfloor 29^{3/4} \rfloor = 12$. But $c \ge 29^{1/4}$, so $\lfloor c^3 \rfloor \ge \lfloor 29^{3/4} \rfloor = 12$ and $\lfloor c^3 \rfloor$ is prime, so $\lfloor c^3 \rfloor \ge 13$ (the next prime after $12$). Thus $c \ge 13^{1/3}$...etc.
Code and computation
sage: def pp(x,m,N):
....: if m==0:
....: return [x,N]
....: n=1
....: while is_prime(floor(x^n)):
....: n+=1
....: y=(next_prime(floor(x^n)))^(1/n)
....: if n>N:
....: print([y.n(digits=50),n])
....: if m>0:
....: return pp(y,m-1,max(N,n))
....:
sage: [x,N]=[2,1]
....: for i in range(50):
....: print([x.n(),i,x])
....: [x,N]=pp(x,2000,N)
....:
[2.00000000000000, 0, 2]
[2.2360679774997896964091736687312762354406183596115, 2]
[2.3205957871060836757951626724361790745664950066620, 4]
[2.3605199968529440968079770148087981739160611080823, 6]
[2.3610399178720974018554873806243419171344471404614, 7]
[2.3614475790074000731522972087149667708348562364648, 8]
[2.3618471924783357334160479093464808403896493219202, 9]
[11.475443618001157866527887755121692154619551039797, 10]
[11.477224946012393993604098825789953501166255954971, 11]
[13.175408468878287715899264969959311747403938659654, 12]
[13.902522805397589015594035741302782303052562472125, 13]
[17.536953877378362827281069096062598297391410332473, 14]
[17.536953877378362887338555293680426557325000127192, 15]
[17.536953877378362887608316800202110377163430446266, 16]
[17.536953877378362888433104776029938028740833100165, 17]
[19.5947182358701, 1, 425844660257^(1/9)]
[23.270857281996347735902084276490188835552979733694, 18]
[23.270857281996347735902086859116418324481035520531, 19]
[23.4071193822604, 2, 300187^(1/4)]
[29.2781355078466, 3, 15808503090647^(1/9)]
[29.283627562413046640114495923799171963094901972525, 20]
[29.283627562413046640114495923799346921502556832750, 22]
[29.283627562413046640114495923800540507023103735417, 23]
[29.3140651944330, 4, 15983962223759^(1/9)]
[29.6863280163833, 5, 468465682453013093^(1/12)]
[29.7199886617202, 6, 608680116727^(1/8)]
[29.7262158681975, 7, 23211229^(1/5)]
[29.7986032159883, 8, 490174696327575403^(1/12)]
[31.1080969548337, 9, 848661378701401^(1/10)]
[31.1656716760603, 10, 815588091830835967331^(1/14)]
[31.3602318150116, 11, 28851733210111963^(1/11)]
[31.4936884203380, 12, 967805781529^(1/8)]
[31.5855793146363, 13, 985975585022214811^(1/12)]
[31.5877240204566, 14, 31239328108632379^(1/11)]
[31.7708284209810, 15, 32673809611^(1/7)]
[31.7776275935340, 16, 1039852569731^(1/8)]
[31.9228719777914, 17, 1141372454772252090287^(1/14)]
[31.9561321469378, 18, 37793885499058215854551493^(1/17)]
[31.956132147003241854271090230723448177838360049016, 24]
[31.956132147003241854271090230723448180095554722710, 25]
[37.0580160390721, 19, 248581122217238450999^(1/13)]
[37.0584389108062, 20, 95986431199^(1/7)]
[37.0658517174746, 21, 96120913169^(1/7)]
[37.1639162040819, 22, 135236290468561^(1/9)]
[37.1669125206849, 23, 9598222132156577814457^(1/14)]
[37.1683288435420, 24, 187026601989858727^(1/11)]
[37.1687268547021, 25, 6952359561958781053^(1/12)]
[37.4113116298099, 26, 281214785630720282953^(1/13)]
[37.4118531804157, 27, 143576377582747^(1/9)]
[37.411951997910355865588788752100197596710593819578, 26]
[37.411951997910355865588788752100197624982207527611, 27]
[37.4133160075168, 28, 5373559002051551^(1/10)]
[37.5366773413800, 29, 147945733482517^(1/9)]
[37.5399875044127, 30, 148063193945647^(1/9)]
[37.5479790983487, 31, 294866460515972260121^(1/13)]
[37.5481223970385, 32, 11072231266616986908067^(1/14)]
[37.7259910967192, 33, 8311662401758930111^(1/12)]
[37.7273014336926, 34, 313707315298504202077^(1/13)]
[37.7315275341604, 35, 8326311451357139347^(1/12)]
[37.7349245483495, 36, 220891164860467163^(1/11)]
[37.7783534119867, 37, 109824990767^(1/7)]
[37.7807097942423, 38, 5925174272790217^(1/10)]
[37.7835286660001, 39, 109930348483^(1/7)]
[37.7863629239001, 40, 320151978004944001769^(1/13)]
[37.8064776911507, 41, 12187848407641642523273^(1/14)]
[37.8562779668531, 42, 12414543823223285559433^(1/14)]
[37.8623613459024, 43, 229236967804694263^(1/11)]
[37.9373990583452, 44, 12792214711968675984467^(1/14)]
[37.9424408570224, 45, 162977402781511^(1/9)]
[37.9426346064883, 46, 234640145197453427^(1/11)]
[37.9444485557837, 47, 163055033768981^(1/9)]
[41.146225319112865984149291850103993604138170925551, 28]
[41.1521261606709, 48, 39947710754575417570267^(1/14)]
[41.1533074278378, 49, 67682425275661104807663881^(1/16)]