Work in progress.  

It is not possible to generate all the numbers with only the give operations.  
I used a computer for that.  
The only ones are `1,2,3,4,6,7,8,9,10,12,15,16,17,19,24,32,48,56,63,64,65,72,80,87,88,89` (marked in bold).  



So I took the liberty to use other functions like square root ($\sqrt{x}$), ceil ($\lceil x \rceil$), floor ($\lfloor x \rfloor$).  

**1**  = $\frac{88}{88}$  
**2**  = $\frac88 + \frac88$  
**3**  = $\frac{88}{8} - 8$  
**4**  = $\frac{8 \times 8}{8+8}$  
5  = $\sqrt{8+8} + \frac88$  
**6**  = $8 - \frac{8 + 8}{8}$  
**7**  = $\frac{8 \times 8 - 8}{8}$  
**8**  = $8 + 8 \times (8 - 8)$    
**9**  = $\frac{8 \times 8 + 8}{8}$  
**10** = $\frac{88 - 8}{8}$  
11 = $\lceil \sqrt8 \rceil + 8-8+8$  
12 = $\frac{88 + 8}{8}$  
13 = $\lfloor \sqrt8 \rfloor + \frac{88}{8}$  
14 = $\lceil \sqrt8 \rceil + \frac{88}{8}$  
**15** = $8 + 8 + \frac88$  
**16** = $8 \times \frac{8+8}{8}$  
**17** = $8 + 8 + \frac88$  
18 = $8 + 8 + \sqrt{\sqrt{8+8}}$  
**19** = $\frac{88}{8} + 8$  
20 = $8 + 8 + \sqrt{8+8}$  
21 = $8 + 8 + \lceil \sqrt{8} \rceil + \lfloor \sqrt{8} \rfloor$  
22 = $\frac{88}{\sqrt{8+8}}$  
23 = $\lceil\sqrt{8}\rceil^{\lceil\sqrt{8}\rceil} - \sqrt{8+8}$  
**24** = $88-8\times 8$  
25 = $\lceil\sqrt{8}\rceil^{\lceil\sqrt{8}\rceil} - \sqrt{\sqrt{8+8}}$  
26 = $ 8 +8 +8 + \lfloor \sqrt{8} \rfloor$   
27 = $\lceil \sqrt8 \rceil + 8+8+8$  
28 = $8 \times \lceil\sqrt8\rceil + \sqrt{8+8}$  
29 = $8 \times \sqrt{8+8} - \lceil \sqrt8 \rceil$   
30 = $8 \times \sqrt{8+8} - \lfloor \sqrt8 \rfloor$    
31 = $\lceil\sqrt{8}\rceil^{\lceil\sqrt{8}\rceil} + \sqrt{8+8}$  
**32** = $8 + 8 + 8 + 8$  
33 = $8 \times \sqrt{8+8} + \lfloor \sqrt{\sqrt8} \rfloor$ 
34 = $8 \times \sqrt{8+8} + \lfloor \sqrt8 \rfloor$    
35 = $8 \times \sqrt{8+8} + \lceil \sqrt8 \rceil$  
36 = $(8 + \lfloor\sqrt{\sqrt8}\rfloor) \times \sqrt{8+8}$  
37 = $\lfloor\frac{88}{\lceil\sqrt8\rceil}\rfloor + 8$  
38 = $\lceil\frac{88}{\lceil\sqrt8\rceil}\rceil + 8$    
39 =  
40 = $8 \times \sqrt{8+8} + 8$  
41 =  
42 =  
43 =  
44 = $\frac{88}{\sqrt{\sqrt{8+8}}}$  
45 =  
46 = $\frac{88}{\lfloor\sqrt8\rfloor} + \lfloor\sqrt8\rfloor$  
47 = $\frac{88}{\lfloor\sqrt8\rfloor} + \lceil\sqrt8\rceil$     
**48**  = $8 \times 8 - 8 -8$  
49 =  
50 =  
51 =  
52 = $\frac{88}{\lfloor\sqrt8\rfloor} + 8$   
53 = $8 \times 8 - 8 - \lceil \sqrt8 \rceil$  
54 = $8 \times 8 - 8 - \lfloor \sqrt8 \rfloor$  
55 =     
**56** = $8 \times (8 - \frac88)$  
57 =   
58 = $8 \times 8 - 8 + \lfloor \sqrt8 \rfloor$    
59 = $8 \times 8 - 8 + \lceil \sqrt8 \rceil$  
60 = $8 \times 8 - \sqrt{8+8}$  
61 = $8\times 8 - \lceil \frac{8}{\sqrt8}\rceil$  
62 = $8\times 8 - \sqrt{\sqrt{8+8}}$  
**63**  = $8 \times 8 + \frac88$   
**64**  = $8 \times 8 \times \frac88$  
**65**  = $8 * 8 + \frac88$    
66 = $8\times 8 + \sqrt{\sqrt{8+8}}$  
67 = $8\times 8 + \lceil \frac{8}{\sqrt8}\rceil$  
68 = $8 \times 8 + \sqrt{8+8}$  
69 = $8 + 8 \times 8 - \lceil \sqrt8 \rceil$  
70 = $8 \times 8 + \sqrt{8+8}$  
71 = $8 +8 \times 8 - \lfloor \sqrt{\sqrt8} \rfloor$  
**72**  = $88 - 8 - 8$  
73 = $\lfloor \sqrt{\sqrt8} \rfloor + 8 +8 \times 8$  
74 = $\lfloor \sqrt8 \rfloor + 8 +8 \times 8$  
75 = $\lceil \sqrt8 \rceil + 8 +8 \times 8$  
76 =   
77 = $88 - 8 - \lceil\sqrt8\rceil$  
78 = $88 - 8 - \lfloor\sqrt8\rfloor$  
79 = $\lceil\sqrt8\rceil^{\sqrt{8+8}} - \lfloor\sqrt8\rfloor$  
**80**  = $8\times 8 + 8 + 8$  
81 = $(\lfloor\sqrt8\rfloor + \lfloor\sqrt{\sqrt8}\rfloor)^{\sqrt{8+8}}$  
82 = $88 - 8 + \lfloor\sqrt8\rfloor$  
83 = $88 - 8 + \lceil\sqrt8\rceil$  
84 = $88 - \sqrt{8+8}$  
85 = $88 - \lceil\frac{8}{\sqrt8}\rceil$  
86 = $88 - \lfloor\frac{8}{\sqrt8}\rfloor$  
**87** = $88 - \frac88$  
**88** = $88 - 8 + 8$  
**89** = $88 + \frac88$  
90 = $88 + \lfloor\frac{8}{\sqrt8}\rfloor$  
91 = $88 + \lceil\frac{8}{\sqrt8}\rceil$  
92 = $88 + \sqrt{8+8}$   
93 = $88 +8 - \lceil\sqrt8\rceil$  
94 = $88 +8 - \lfloor\sqrt8\rfloor$  
95 = $88 +8 - \lfloor\sqrt{\sqrt8}\rfloor$    
96 = $8 \times \sqrt{8+8} \times \lceil \sqrt8 \rceil$   
97 = $88 +8 + \lfloor\sqrt{\sqrt8}\rfloor$   
98 = $88 +8 + \lfloor\sqrt8\rfloor$  
99 = $88 +8 + \lceil\sqrt8\rceil$  
100 =