Consider the DE:
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$\mathrm{DE}\u2254\mathrm{diff}\left(y\left(t\right)\,t\right)=\mathrm{cos}\left({t}^{2}\right)y\left(t\right)$

${\mathrm{DE}}{\u2254}\frac{{\ⅆ}}{{\ⅆ}{t}}\phantom{\rule[0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){=}{\mathrm{cos}}{}\left({{t}}^{{2}}\right){}{y}{}\left({t}\right)$
 (1) 
A numerical routine for the solution of this DE is obtained by using dsolve and the rkf45 method as follows:
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$\mathrm{ysol}\u2254\mathrm{dsolve}\left(\left\{\mathrm{DE}\,y\left(0\right)=1\right\}\,\mathrm{numeric}\,\mathrm{method}=\mathrm{rkf45}\right)$

${\mathrm{ysol}}{\u2254}{\mathbf{proc}}\left({\mathrm{x\_rkf45}}\right)\phantom{\rule[0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[0.0ex]{0.5em}{0.0ex}}{\mathbf{end\; proc}}$
 (2) 
The routine is then applied to a value to obtain the numerical solution at that point:
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$\mathrm{ysol}\left(5\right)$

$\left[{t}{=}{5.}{\,}{y}{}\left({t}\right){=}{1.84313059855996}\right]$
 (3) 
Suppose that instead we wanted to integrate out to $t=200$. If we attempt to do this with the generated routine, a problem occurs.
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$\mathrm{ysol}\left(200\right)$

Integrating this DE out to $t=200$ actually requires nearly half a million function evaluations due to the $\mathrm{O}\left({t}^{2}\right)$ increase in the rate of change of the function on the righthand side. This can take some time for computation, so maxfun acts like a stopcheck here.
If we really want this value, we can increase the value of maxfun from its default of $30000$ to $500000$, and obtain a solution as follows:
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$\mathrm{ysol}\u2254\mathrm{dsolve}\left(\left\{\mathrm{DE}\,y\left(0\right)=1\right\}\,\mathrm{numeric}\,\mathrm{method}=\mathrm{rkf45}\,\mathrm{maxfun}=500000\right)$

${\mathrm{ysol}}{\u2254}{\mathbf{proc}}\left({\mathrm{x\_rkf45}}\right)\phantom{\rule[0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[0.0ex]{0.5em}{0.0ex}}{\mathbf{end\; proc}}$
 (4) 
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$\mathrm{ysol}\left(200\right)$

$\left[{t}{=}{200.}{\,}{y}{}\left({t}\right){=}{1.87563347676955}\right]$
 (5) 
Alternatively, we could have set maxfun to $0$, and then the routine would proceed until it found the value at the specified point, or recognized a singularity.
The ODE IVP:
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$\mathrm{dsys}\u2254\left\{\mathrm{diff}\left(y\left(t\right)\,t\right)=\frac{{y\left(t\right)}^{2}}{1100y\left(t\right)}\,y\left(0\right)=\frac{1}{200}\right\}\:$

has a fast blowup singularity just past $t=30$.
If we attempt to numerically integrate this IVP to $t=31$, we get:
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$\mathrm{ysol}\u2254\mathrm{dsolve}\left(\mathrm{dsys}\,\mathrm{numeric}\,\mathrm{method}=\mathrm{rkf45}\,\mathrm{abserr}=1.\times {10}^{\mathrm{7}}\,\mathrm{relerr}=1.\times {10}^{\mathrm{7}}\right)$

${\mathrm{ysol}}{\u2254}{\mathbf{proc}}\left({\mathrm{x\_rkf45}}\right)\phantom{\rule[0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[0.0ex]{0.5em}{0.0ex}}{\mathbf{end\; proc}}$
 (6) 
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$\mathrm{ysol}\left(31\right)$

Or implicitly using the range argument:
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$\mathrm{ysol}\u2254\mathrm{dsolve}\left(\mathrm{dsys}\,\mathrm{numeric}\,\mathrm{method}=\mathrm{rkf45}\,\mathrm{abserr}=1.\times {10}^{\mathrm{7}}\,\mathrm{relerr}=1.\times {10}^{\mathrm{7}}\,\mathrm{range}=0..31\right)$

${\mathrm{ysol}}{\u2254}{\mathbf{proc}}\left({\mathrm{x\_rkf45}}\right)\phantom{\rule[0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[0.0ex]{0.5em}{0.0ex}}{\mathbf{end\; proc}}$
 (7) 
This error tells us that integration over this region required more than the default of $30000$ evaluations of the DE. This seems to indicate that we need to increase maxfun. In doing so, we get:
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$\mathrm{ysol}\u2254\mathrm{dsolve}\left(\mathrm{dsys}\,\mathrm{numeric}\,\mathrm{method}=\mathrm{rkf45}\,\mathrm{abserr}=1.\times {10}^{\mathrm{7}}\,\mathrm{relerr}=1.\times {10}^{\mathrm{7}}\,\mathrm{maxfun}=100000\right)$

${\mathrm{ysol}}{\u2254}{\mathbf{proc}}\left({\mathrm{x\_rkf45}}\right)\phantom{\rule[0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[0.0ex]{0.5em}{0.0ex}}{\mathbf{end\; proc}}$
 (8) 
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$\mathrm{ysol}\left(31\right)$

we see that we still hit the maxfun barrier, though the integration does proceed a small bit further.
Since there is an actual singularity, this is a case where the singularity detection fails and maxfun set to $30000$ prevents the calculation from running for an excessive time while serving no useful purpose.